forked from personal/squiggle.c
9034 lines
581 KiB
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9034 lines
581 KiB
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<title>Gamma distribution - Wikipedia</title>
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|
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<li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Gamma+distribution" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a>
|
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</li>
|
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<li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Gamma+distribution" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a>
|
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|
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|
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|
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|
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|
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<input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" >
|
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<label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span>
|
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|
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<span class="vector-dropdown-label-text">Personal tools</span>
|
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|
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|
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|
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|
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|
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|
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|
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<li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Gamma+distribution" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Gamma+distribution" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li>
|
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|
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|
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|
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|
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</div>
|
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|
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|
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<div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" >
|
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|
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|
|
Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a>
|
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|
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|
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|
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|
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<ul class="vector-menu-content-list">
|
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|
|||
|
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<li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li>
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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<div class="mw-page-container-inner">
|
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|
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|
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<div id="siteNotice"><!-- CentralNotice --></div>
|
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|
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<div id="mw-navigation">
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<nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site" role="navigation">
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<div id="vector-main-menu-pinned-container" class="vector-pinned-container">
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|
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|
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|
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|
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<div class="vector-sticky-pinned-container">
|
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|
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<nav id="mw-panel-toc" role="navigation" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark">
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<div id="vector-toc-pinned-container" class="vector-pinned-container">
|
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<div id="vector-toc" class="vector-toc vector-pinnable-element">
|
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<div
|
|||
|
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class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned"
|
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|
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data-feature-name="toc-pinned"
|
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|
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|
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|
|||
|
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|
|||
|
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>
|
|||
|
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<h2 class="vector-pinnable-header-label">Contents</h2>
|
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|
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<button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button>
|
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<button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button>
|
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|
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</div>
|
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|
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|
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|
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<ul class="vector-toc-contents" id="mw-panel-toc-list">
|
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<li id="toc-mw-content-text"
|
|||
|
|
class="vector-toc-list-item vector-toc-level-1">
|
|||
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<a href="#" class="vector-toc-link">
|
|||
|
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<div class="vector-toc-text">(Top)</div>
|
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|
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</a>
|
|||
|
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</li>
|
|||
|
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<li id="toc-Definitions"
|
|||
|
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class="vector-toc-list-item vector-toc-level-1">
|
|||
|
|
<a class="vector-toc-link" href="#Definitions">
|
|||
|
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<div class="vector-toc-text">
|
|||
|
|
<span class="vector-toc-numb">1</span>Definitions</div>
|
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|
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</a>
|
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|
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|
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<button aria-controls="toc-Definitions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle">
|
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|
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<span class="vector-icon vector-icon--x-small mw-ui-icon-wikimedia-expand"></span>
|
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|
|
<span>Toggle Definitions subsection</span>
|
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|
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</button>
|
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|
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|
|
<ul id="toc-Definitions-sublist" class="vector-toc-list">
|
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<li id="toc-Characterization_using_shape_α_and_rate_β"
|
|||
|
|
class="vector-toc-list-item vector-toc-level-2">
|
|||
|
|
<a class="vector-toc-link" href="#Characterization_using_shape_α_and_rate_β">
|
|||
|
|
<div class="vector-toc-text">
|
|||
|
|
<span class="vector-toc-numb">1.1</span>Characterization using shape <i>α</i> and rate <i>β</i></div>
|
|||
|
|
</a>
|
|||
|
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|
|||
|
|
<ul id="toc-Characterization_using_shape_α_and_rate_β-sublist" class="vector-toc-list">
|
|||
|
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</ul>
|
|||
|
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</li>
|
|||
|
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<li id="toc-Characterization_using_shape_k_and_scale_θ"
|
|||
|
|
class="vector-toc-list-item vector-toc-level-2">
|
|||
|
|
<a class="vector-toc-link" href="#Characterization_using_shape_k_and_scale_θ">
|
|||
|
|
<div class="vector-toc-text">
|
|||
|
|
<span class="vector-toc-numb">1.2</span>Characterization using shape <i>k</i> and scale <i>θ</i></div>
|
|||
|
|
</a>
|
|||
|
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|
|||
|
|
<ul id="toc-Characterization_using_shape_k_and_scale_θ-sublist" class="vector-toc-list">
|
|||
|
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</ul>
|
|||
|
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</li>
|
|||
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</ul>
|
|||
|
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</li>
|
|||
|
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<li id="toc-Properties"
|
|||
|
|
class="vector-toc-list-item vector-toc-level-1">
|
|||
|
|
<a class="vector-toc-link" href="#Properties">
|
|||
|
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<div class="vector-toc-text">
|
|||
|
|
<span class="vector-toc-numb">2</span>Properties</div>
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
<button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle">
|
|||
|
|
<span class="vector-icon vector-icon--x-small mw-ui-icon-wikimedia-expand"></span>
|
|||
|
|
<span>Toggle Properties subsection</span>
|
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|
|
</button>
|
|||
|
|
|
|||
|
|
<ul id="toc-Properties-sublist" class="vector-toc-list">
|
|||
|
|
<li id="toc-Mean_and_variance"
|
|||
|
|
class="vector-toc-list-item vector-toc-level-2">
|
|||
|
|
<a class="vector-toc-link" href="#Mean_and_variance">
|
|||
|
|
<div class="vector-toc-text">
|
|||
|
|
<span class="vector-toc-numb">2.1</span>Mean and variance</div>
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
<ul id="toc-Mean_and_variance-sublist" class="vector-toc-list">
|
|||
|
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</ul>
|
|||
|
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</li>
|
|||
|
|
<li id="toc-Skewness"
|
|||
|
|
class="vector-toc-list-item vector-toc-level-2">
|
|||
|
|
<a class="vector-toc-link" href="#Skewness">
|
|||
|
|
<div class="vector-toc-text">
|
|||
|
|
<span class="vector-toc-numb">2.2</span>Skewness</div>
|
|||
|
|
</a>
|
|||
|
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|
|||
|
|
<ul id="toc-Skewness-sublist" class="vector-toc-list">
|
|||
|
|
</ul>
|
|||
|
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</li>
|
|||
|
|
<li id="toc-Higher_moments"
|
|||
|
|
class="vector-toc-list-item vector-toc-level-2">
|
|||
|
|
<a class="vector-toc-link" href="#Higher_moments">
|
|||
|
|
<div class="vector-toc-text">
|
|||
|
|
<span class="vector-toc-numb">2.3</span>Higher moments</div>
|
|||
|
|
</a>
|
|||
|
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|
|||
|
|
<ul id="toc-Higher_moments-sublist" class="vector-toc-list">
|
|||
|
|
</ul>
|
|||
|
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</li>
|
|||
|
|
<li id="toc-Median_approximations_and_bounds"
|
|||
|
|
class="vector-toc-list-item vector-toc-level-2">
|
|||
|
|
<a class="vector-toc-link" href="#Median_approximations_and_bounds">
|
|||
|
|
<div class="vector-toc-text">
|
|||
|
|
<span class="vector-toc-numb">2.4</span>Median approximations and bounds</div>
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
<ul id="toc-Median_approximations_and_bounds-sublist" class="vector-toc-list">
|
|||
|
|
</ul>
|
|||
|
|
</li>
|
|||
|
|
<li id="toc-Summation"
|
|||
|
|
class="vector-toc-list-item vector-toc-level-2">
|
|||
|
|
<a class="vector-toc-link" href="#Summation">
|
|||
|
|
<div class="vector-toc-text">
|
|||
|
|
<span class="vector-toc-numb">2.5</span>Summation</div>
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
<ul id="toc-Summation-sublist" class="vector-toc-list">
|
|||
|
|
</ul>
|
|||
|
|
</li>
|
|||
|
|
<li id="toc-Scaling"
|
|||
|
|
class="vector-toc-list-item vector-toc-level-2">
|
|||
|
|
<a class="vector-toc-link" href="#Scaling">
|
|||
|
|
<div class="vector-toc-text">
|
|||
|
|
<span class="vector-toc-numb">2.6</span>Scaling</div>
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
<ul id="toc-Scaling-sublist" class="vector-toc-list">
|
|||
|
|
</ul>
|
|||
|
|
</li>
|
|||
|
|
<li id="toc-Exponential_family"
|
|||
|
|
class="vector-toc-list-item vector-toc-level-2">
|
|||
|
|
<a class="vector-toc-link" href="#Exponential_family">
|
|||
|
|
<div class="vector-toc-text">
|
|||
|
|
<span class="vector-toc-numb">2.7</span>Exponential family</div>
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
<ul id="toc-Exponential_family-sublist" class="vector-toc-list">
|
|||
|
|
</ul>
|
|||
|
|
</li>
|
|||
|
|
<li id="toc-Logarithmic_expectation_and_variance"
|
|||
|
|
class="vector-toc-list-item vector-toc-level-2">
|
|||
|
|
<a class="vector-toc-link" href="#Logarithmic_expectation_and_variance">
|
|||
|
|
<div class="vector-toc-text">
|
|||
|
|
<span class="vector-toc-numb">2.8</span>Logarithmic expectation and variance</div>
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
<ul id="toc-Logarithmic_expectation_and_variance-sublist" class="vector-toc-list">
|
|||
|
|
</ul>
|
|||
|
|
</li>
|
|||
|
|
<li id="toc-Information_entropy"
|
|||
|
|
class="vector-toc-list-item vector-toc-level-2">
|
|||
|
|
<a class="vector-toc-link" href="#Information_entropy">
|
|||
|
|
<div class="vector-toc-text">
|
|||
|
|
<span class="vector-toc-numb">2.9</span>Information entropy</div>
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
<ul id="toc-Information_entropy-sublist" class="vector-toc-list">
|
|||
|
|
</ul>
|
|||
|
|
</li>
|
|||
|
|
<li id="toc-Kullback–Leibler_divergence"
|
|||
|
|
class="vector-toc-list-item vector-toc-level-2">
|
|||
|
|
<a class="vector-toc-link" href="#Kullback–Leibler_divergence">
|
|||
|
|
<div class="vector-toc-text">
|
|||
|
|
<span class="vector-toc-numb">2.10</span>Kullback–Leibler divergence</div>
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
<ul id="toc-Kullback–Leibler_divergence-sublist" class="vector-toc-list">
|
|||
|
|
</ul>
|
|||
|
|
</li>
|
|||
|
|
<li id="toc-Laplace_transform"
|
|||
|
|
class="vector-toc-list-item vector-toc-level-2">
|
|||
|
|
<a class="vector-toc-link" href="#Laplace_transform">
|
|||
|
|
<div class="vector-toc-text">
|
|||
|
|
<span class="vector-toc-numb">2.11</span>Laplace transform</div>
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
<ul id="toc-Laplace_transform-sublist" class="vector-toc-list">
|
|||
|
|
</ul>
|
|||
|
|
</li>
|
|||
|
|
</ul>
|
|||
|
|
</li>
|
|||
|
|
<li id="toc-Related_distributions"
|
|||
|
|
class="vector-toc-list-item vector-toc-level-1">
|
|||
|
|
<a class="vector-toc-link" href="#Related_distributions">
|
|||
|
|
<div class="vector-toc-text">
|
|||
|
|
<span class="vector-toc-numb">3</span>Related distributions</div>
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
<button aria-controls="toc-Related_distributions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle">
|
|||
|
|
<span class="vector-icon vector-icon--x-small mw-ui-icon-wikimedia-expand"></span>
|
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|
|
<span>Toggle Related distributions subsection</span>
|
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|
|
</button>
|
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|
|
|
|||
|
|
<ul id="toc-Related_distributions-sublist" class="vector-toc-list">
|
|||
|
|
<li id="toc-General"
|
|||
|
|
class="vector-toc-list-item vector-toc-level-2">
|
|||
|
|
<a class="vector-toc-link" href="#General">
|
|||
|
|
<div class="vector-toc-text">
|
|||
|
|
<span class="vector-toc-numb">3.1</span>General</div>
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
<ul id="toc-General-sublist" class="vector-toc-list">
|
|||
|
|
</ul>
|
|||
|
|
</li>
|
|||
|
|
<li id="toc-Compound_gamma"
|
|||
|
|
class="vector-toc-list-item vector-toc-level-2">
|
|||
|
|
<a class="vector-toc-link" href="#Compound_gamma">
|
|||
|
|
<div class="vector-toc-text">
|
|||
|
|
<span class="vector-toc-numb">3.2</span>Compound gamma</div>
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
<ul id="toc-Compound_gamma-sublist" class="vector-toc-list">
|
|||
|
|
</ul>
|
|||
|
|
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|
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|
|
<li id="toc-Weibull_and_stable_count"
|
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<span class="vector-toc-numb">3.3</span>Weibull and stable count</div>
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<span class="vector-toc-numb">4</span>Statistical inference</div>
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<span class="vector-toc-numb">4.1</span>Parameter estimation</div>
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<span class="vector-toc-numb">4.1.1</span>Maximum likelihood estimation</div>
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<span class="vector-toc-numb">4.1.1.1</span>Caveat for small shape parameter</div>
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<span class="vector-toc-numb">4.1.2</span>Closed-form estimators</div>
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<span class="vector-toc-numb">4.1.3</span>Bayesian minimum mean squared error</div>
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<span class="vector-toc-numb">4.2</span>Bayesian inference</div>
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<span class="vector-toc-numb">4.2.1</span>Conjugate prior</div>
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<a class="vector-toc-link" href="#Occurrence_and_applications">
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<div class="vector-toc-text">
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<span class="vector-toc-numb">5</span>Occurrence and applications</div>
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|
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<div class="vector-toc-text">
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<span class="vector-toc-numb">6</span>Random variate generation</div>
|
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<span class="vector-toc-numb">7</span>References</div>
|
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|
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<span class="vector-toc-numb">8</span>External links</div>
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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Gamma distribution</span></h1>
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<li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%88%D8%B2%D9%8A%D8%B9_%D8%BA%D8%A7%D9%85%D8%A7" title="توزيع غاما – Arabic" lang="ar" hreflang="ar" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Qamma_paylanmas%C4%B1" title="Qamma paylanması – Azerbaijani" lang="az" hreflang="az" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A6%BE%E0%A6%AE%E0%A6%BE_%E0%A6%AC%E0%A6%A3%E0%A7%8D%E0%A6%9F%E0%A6%A8" title="গামা বণ্টন – Bangla" lang="bn" hreflang="bn" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%93%D0%B0%D0%BC%D0%B0-%D1%80%D0%B0%D0%B7%D0%BC%D0%B5%D1%80%D0%BA%D0%B0%D0%B2%D0%B0%D0%BD%D0%BD%D0%B5" title="Гама-размеркаванне – Belarusian" lang="be" hreflang="be" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Distribuci%C3%B3_gamma" title="Distribució gamma – Catalan" lang="ca" hreflang="ca" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Rozd%C4%9Blen%C3%AD_gama" title="Rozdělení gama – Czech" lang="cs" hreflang="cs" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Gammaverteilung" title="Gammaverteilung – German" lang="de" hreflang="de" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Distribuci%C3%B3n_gamma" title="Distribución gamma – Spanish" lang="es" hreflang="es" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%88%D8%B2%DB%8C%D8%B9_%DA%AF%D8%A7%D9%85%D8%A7" title="توزیع گاما – Persian" lang="fa" hreflang="fa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Loi_Gamma" title="Loi Gamma – French" lang="fr" hreflang="fr" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Distribuci%C3%B3n_gamma" title="Distribución gamma – Galician" lang="gl" hreflang="gl" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B0%90%EB%A7%88_%EB%B6%84%ED%8F%AC" title="감마 분포 – Korean" lang="ko" hreflang="ko" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Distribuzione_Gamma" title="Distribuzione Gamma – Italian" lang="it" hreflang="it" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%AA%D7%A4%D7%9C%D7%92%D7%95%D7%AA_%D7%92%D7%9E%D7%90" title="התפלגות גמא – Hebrew" lang="he" hreflang="he" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Gamma-eloszl%C3%A1s" title="Gamma-eloszlás – Hungarian" lang="hu" hreflang="hu" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Gamma-v
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<style data-mw-deduplicate="TemplateStyles:r1097763485">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}html.client-js body.skin-minerva .mw-parser-output .mbox-text-span{margin-left:23px!important}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}</style><table class="box-Lead_too_short plainlinks metadata ambox ambox-content ambox-lead_too_short" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/6/6c/Wiki_letter_w.svg/40px-Wiki_letter_w.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/6/6c/Wiki_letter_w.svg/60px-Wiki_letter_w.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/6/6c/Wiki_letter_w.svg/80px-Wiki_letter_w.svg.png 2x" data-file-width="44" data-file-height="44" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article's <a href="/wiki/Wikipedia:Manual_of_Style/Lead_section#Length" title="Wikipedia:Manual of Style/Lead section">lead section</a> <b>may be too short to adequately <a href="/wiki/Wikipedia:Summary_style" title="Wikipedia:Summary style">summarize</a> the key points</b>.<span class="hide-when-compact"> Please consider expanding the lead to <a href="/wiki/Wikipedia:Manual_of_Style/Lead_section#Provide_an_accessible_overview" title="Wikipedia:Manual of Style/Lead section">provide an accessible overview</a> of all important aspects of the article.</span> <span class="date-container"><i>(<span class="date">November 2022</span>)</i></span></div></td></tr></tbody></table>
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<style data-mw-deduplicate="TemplateStyles:r1066479718">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}body.skin-minerva .mw-parser-output .infobox-header,body.skin-minerva .mw-parser-output .infobox-subheader,body.skin-minerva .mw-parser-output .infobox-above,body.skin-minerva .mw-parser-output .infobox-title,body.skin-minerva .mw-parser-output .infobox-image,body.skin-minerva .mw-parser-output .infobox-full-data,body.skin-minerva .mw-parser-output .infobox-below{text-align:center}</style><style data-mw-deduplicate="TemplateStyles:r1046248152">.mw-parser-output .ib-prob-dist{border-collapse:collapse;width:20em}.mw-parser-output .ib-prob-dist td,.mw-parser-output .ib-prob-dist th{border:1px solid #a2a9b1}.mw-parser-output .ib-prob-dist .infobox-subheader{text-align:left}.mw-parser-output .ib-prob-dist-image{background:#ddd;font-weight:bold;text-align:center}</style><table class="infobox ib-prob-dist"><caption class="infobox-title">Gamma</caption><tbody><tr><td colspan="4" class="infobox-image">
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<div class="ib-prob-dist-image">Probability density function</div><span typeof="mw:File"><a href="/wiki/File:Gamma_distribution_pdf.svg" class="mw-file-description" title="Probability density plots of gamma distributions"><img alt="Probability density plots of gamma distributions" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Gamma_distribution_pdf.svg/325px-Gamma_distribution_pdf.svg.png" decoding="async" width="325" height="244" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Gamma_distribution_pdf.svg/488px-Gamma_distribution_pdf.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Gamma_distribution_pdf.svg/650px-Gamma_distribution_pdf.svg.png 2x" data-file-width="800" data-file-height="600" /></a></span></td></tr><tr><td colspan="4" class="infobox-image">
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<div class="ib-prob-dist-image">Cumulative distribution function</div><span typeof="mw:File"><a href="/wiki/File:Gamma_distribution_cdf.svg" class="mw-file-description" title="Cumulative distribution plots of gamma distributions"><img alt="Cumulative distribution plots of gamma distributions" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Gamma_distribution_cdf.svg/325px-Gamma_distribution_cdf.svg.png" decoding="async" width="325" height="244" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Gamma_distribution_cdf.svg/488px-Gamma_distribution_cdf.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Gamma_distribution_cdf.svg/650px-Gamma_distribution_cdf.svg.png 2x" data-file-width="800" data-file-height="600" /></a></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameters</a></th><td class="infobox-data infobox-data-a">
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<ul><li><i>k</i> > 0 <a href="/wiki/Shape_parameter" title="Shape parameter">shape</a></li>
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<div><ul><li><i>α</i> > 0 <a href="/wiki/Shape_parameter" title="Shape parameter">shape</a></li><li><i>β</i> > 0 <a href="/wiki/Rate_parameter" class="mw-redirect" title="Rate parameter">rate</a></li></ul></div></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Support_(mathematics)" title="Support (mathematics)">Support</a></th><td class="infobox-data infobox-data-a">
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<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {1}{\Gamma (k)\theta ^{k}}}x^{k-1}e^{-x/\theta }}">
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<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<msup>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<msup>
|
|||
|
|
<mi>e</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>/</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {1}{\Gamma (k)\theta ^{k}}}x^{k-1}e^{-x/\theta }}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/959c7540287d0d1be72c2e39826eaf2060ce907e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.651ex; height:6.009ex;" alt="{\displaystyle f(x)={\frac {1}{\Gamma (k)\theta ^{k}}}x^{k-1}e^{-x/\theta }}"></span></td><td class="infobox-data infobox-data-b">
|
|||
|
|
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>f</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<msup>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mrow>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<msup>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<msup>
|
|||
|
|
<mi>e</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3373bd46a7261833556cec0a5c95f51ab4b0e74d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.293ex; height:6.176ex;" alt="{\displaystyle f(x)={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">CDF</a></th><td class="infobox-data infobox-data-a">
|
|||
|
|
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)={\frac {1}{\Gamma (k)}}\gamma \left(k,{\frac {x}{\theta }}\right)}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>F</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mrow>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>γ<!-- γ --></mi>
|
|||
|
|
<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle F(x)={\frac {1}{\Gamma (k)}}\gamma \left(k,{\frac {x}{\theta }}\right)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/206171e583bb2aa2e14a65530b7cd9e5596d922e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.124ex; height:6.009ex;" alt="{\displaystyle F(x)={\frac {1}{\Gamma (k)}}\gamma \left(k,{\frac {x}{\theta }}\right)}"></span></td><td class="infobox-data infobox-data-b">
|
|||
|
|
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)={\frac {1}{\Gamma (\alpha )}}\gamma (\alpha ,\beta x)}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>F</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mrow>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>γ<!-- γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle F(x)={\frac {1}{\Gamma (\alpha )}}\gamma (\alpha ,\beta x)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cd8662ff6fdff9e9f96399ac86737471db7d976" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.819ex; height:6.009ex;" alt="{\displaystyle F(x)={\frac {1}{\Gamma (\alpha )}}\gamma (\alpha ,\beta x)}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Expected_value" title="Expected value">Mean</a></th><td class="infobox-data infobox-data-a">
|
|||
|
|
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\theta }">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k\theta }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/128f9dbb916e39f620213e206592bf14f13554d2" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.302ex; height:2.176ex;" alt="{\displaystyle k\theta }"></span></td><td class="infobox-data infobox-data-b">
|
|||
|
|
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha }{\beta }}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha }{\beta }}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e9485a5ec6938007ea1b5f5f09946f39a14a42f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:2.324ex; height:5.176ex;" alt="{\frac {\alpha }{\beta }}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Median" title="Median">Median</a></th><td class="infobox-data infobox-data-a">
|
|||
|
|
No simple closed form</td><td class="infobox-data infobox-data-b">
|
|||
|
|
No simple closed form</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Mode_(statistics)" title="Mode (statistics)">Mode</a></th><td class="infobox-data infobox-data-a">
|
|||
|
|
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k-1)\theta {\text{ for }}k\geq 1}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mtext> for </mtext>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>≥<!-- ≥ --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle (k-1)\theta {\text{ for }}k\geq 1}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f9e723ed5d8686d56eda7a66c3f1a1fe3db880f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.533ex; height:2.843ex;" alt="{\displaystyle (k-1)\theta {\text{ for }}k\geq 1}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0{\text{ for }}k<1}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mn>0</mn>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mtext> for </mtext>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo><</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle 0{\text{ for }}k<1}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f83ac368070086266d1302f673abb085ec278d46" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.582ex; height:2.176ex;" alt="{\displaystyle 0{\text{ for }}k<1}"></span></td><td class="infobox-data infobox-data-b">
|
|||
|
|
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha -1}{\beta }}{\text{ for }}\alpha \geq 1{\text{, }}0{\text{ for }}\alpha <1}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mtext> for </mtext>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo>≥<!-- ≥ --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mtext>, </mtext>
|
|||
|
|
</mrow>
|
|||
|
|
<mn>0</mn>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mtext> for </mtext>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo><</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha -1}{\beta }}{\text{ for }}\alpha \geq 1{\text{, }}0{\text{ for }}\alpha <1}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0209d0c976e5707524172b0d4071a3a73ad6f12" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.108ex; height:5.676ex;" alt="{\displaystyle {\frac {\alpha -1}{\beta }}{\text{ for }}\alpha \geq 1{\text{, }}0{\text{ for }}\alpha <1}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Variance" title="Variance">Variance</a></th><td class="infobox-data infobox-data-a">
|
|||
|
|
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\theta ^{2}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<msup>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k\theta ^{2}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc9fb845e6076205ac8deabbbc533acedc15103d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.356ex; height:2.676ex;" alt="{\displaystyle k\theta ^{2}}"></span></td><td class="infobox-data infobox-data-b">
|
|||
|
|
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha }{\beta ^{2}}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<msup>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha }{\beta ^{2}}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84135b4731462ff6c9a9c23a434a06e9305ee85e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:3.227ex; height:5.343ex;" alt="{\displaystyle {\frac {\alpha }{\beta ^{2}}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Skewness" title="Skewness">Skewness</a></th><td class="infobox-data infobox-data-a">
|
|||
|
|
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2}{\sqrt {k}}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
<msqrt>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</msqrt>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\frac {2}{\sqrt {k}}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45eff79aec609af84bf67d9272aebc1fa8681994" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:3.983ex; height:6.176ex;" alt="{\displaystyle {\frac {2}{\sqrt {k}}}}"></span></td><td class="infobox-data infobox-data-b">
|
|||
|
|
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2}{\sqrt {\alpha }}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
<msqrt>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
</msqrt>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\frac {2}{\sqrt {\alpha }}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd8a164ebce39b73870af9d23275a9a76c15e6eb" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:4.26ex; height:6.176ex;" alt="{\displaystyle {\frac {2}{\sqrt {\alpha }}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Excess_kurtosis" class="mw-redirect" title="Excess kurtosis">Ex. kurtosis</a></th><td class="infobox-data infobox-data-a">
|
|||
|
|
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {6}{k}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>6</mn>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\frac {6}{k}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bea39cf9fc4c3ff5dc63035a85205b437db7e426" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.047ex; height:5.343ex;" alt="{\displaystyle {\frac {6}{k}}}"></span></td><td class="infobox-data infobox-data-b">
|
|||
|
|
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {6}{\alpha }}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>6</mn>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\frac {6}{\alpha }}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2ce5c506f119628c417e693b0368db51ee874ec" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.324ex; height:5.176ex;" alt="{\displaystyle {\frac {6}{\alpha }}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Information_entropy" class="mw-redirect" title="Information entropy">Entropy</a></th><td class="infobox-data infobox-data-a">
|
|||
|
|
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}k&+\ln \theta +\ln \Gamma (k)\\&+(1-k)\psi (k)\end{aligned}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
|
|||
|
|
<mtr>
|
|||
|
|
<mtd>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mtd>
|
|||
|
|
<mtd>
|
|||
|
|
<mi></mi>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mtd>
|
|||
|
|
</mtr>
|
|||
|
|
<mtr>
|
|||
|
|
<mtd />
|
|||
|
|
<mtd>
|
|||
|
|
<mi></mi>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mi>ψ<!-- ψ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mtd>
|
|||
|
|
</mtr>
|
|||
|
|
</mtable>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}k&+\ln \theta +\ln \Gamma (k)\\&+(1-k)\psi (k)\end{aligned}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/badcd9de56aee1547b1743cd802484471cf3555c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.86ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}k&+\ln \theta +\ln \Gamma (k)\\&+(1-k)\psi (k)\end{aligned}}}"></span></td><td class="infobox-data infobox-data-b">
|
|||
|
|
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\alpha &-\ln \beta +\ln \Gamma (\alpha )\\&+(1-\alpha )\psi (\alpha )\end{aligned}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
|
|||
|
|
<mtr>
|
|||
|
|
<mtd>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
</mtd>
|
|||
|
|
<mtd>
|
|||
|
|
<mi></mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mtd>
|
|||
|
|
</mtr>
|
|||
|
|
<mtr>
|
|||
|
|
<mtd />
|
|||
|
|
<mtd>
|
|||
|
|
<mi></mi>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mi>ψ<!-- ψ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mtd>
|
|||
|
|
</mtr>
|
|||
|
|
</mtable>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\alpha &-\ln \beta +\ln \Gamma (\alpha )\\&+(1-\alpha )\psi (\alpha )\end{aligned}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1a73a699028dd881daebebc59d36f894e74f187" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.655ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\alpha &-\ln \beta +\ln \Gamma (\alpha )\\&+(1-\alpha )\psi (\alpha )\end{aligned}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Moment-generating_function" title="Moment-generating function">MGF</a></th><td class="infobox-data infobox-data-a">
|
|||
|
|
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-\theta t)^{-k}{\text{ for }}t<{\frac {1}{\theta }}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mi>t</mi>
|
|||
|
|
<msup>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mtext> for </mtext>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>t</mi>
|
|||
|
|
<mo><</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle (1-\theta t)^{-k}{\text{ for }}t<{\frac {1}{\theta }}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48369eff20942cb3a2e4cfbef6c31d75c2fca94b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.993ex; height:5.343ex;" alt="{\displaystyle (1-\theta t)^{-k}{\text{ for }}t<{\frac {1}{\theta }}}"></span></td><td class="infobox-data infobox-data-b">
|
|||
|
|
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(1-{\frac {t}{\beta }}\right)^{-\alpha }{\text{ for }}t<\beta }">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<msup>
|
|||
|
|
<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<mrow>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mi>t</mi>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mtext> for </mtext>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>t</mi>
|
|||
|
|
<mo><</mo>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \left(1-{\frac {t}{\beta }}\right)^{-\alpha }{\text{ for }}t<\beta }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae293b4a1441aae3cafacb8a4ce61d23bfa48cb" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.372ex; height:6.509ex;" alt="{\displaystyle \left(1-{\frac {t}{\beta }}\right)^{-\alpha }{\text{ for }}t<\beta }"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">CF</a></th><td class="infobox-data infobox-data-a">
|
|||
|
|
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-\theta it)^{-k}}">
|
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|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
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<mstyle displaystyle="true" scriptlevel="0">
|
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<mo stretchy="false">(</mo>
|
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|
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<mn>1</mn>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
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<mi>i</mi>
|
|||
|
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<mi>t</mi>
|
|||
|
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<msup>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
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<mo>−<!-- − --></mo>
|
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<mi>k</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle (1-\theta it)^{-k}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5768a597d0f48f216a027bd3e92a8b3e67e0371b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.912ex; height:3.176ex;" alt="{\displaystyle (1-\theta it)^{-k}}"></span></td><td class="infobox-data infobox-data-b">
|
|||
|
|
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(1-{\frac {it}{\beta }}\right)^{-\alpha }}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
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<mstyle displaystyle="true" scriptlevel="0">
|
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<msup>
|
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<mrow>
|
|||
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<mo>(</mo>
|
|||
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<mrow>
|
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|
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<mn>1</mn>
|
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<mo>−<!-- − --></mo>
|
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<mrow class="MJX-TeXAtom-ORD">
|
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<mfrac>
|
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|
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<mi>i</mi>
|
|||
|
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<mi>t</mi>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
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<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
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<mrow class="MJX-TeXAtom-ORD">
|
|||
|
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<mo>−<!-- − --></mo>
|
|||
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<mi>α<!-- α --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \left(1-{\frac {it}{\beta }}\right)^{-\alpha }}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e9b1c060b013f2c7f698d58711983d9ddf0b18f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.465ex; height:6.509ex;" alt="{\displaystyle \left(1-{\frac {it}{\beta }}\right)^{-\alpha }}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Fisher_information" title="Fisher information">Fisher information</a></th><td class="infobox-data infobox-data-a">
|
|||
|
|
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(k,\theta )={\begin{pmatrix}\psi ^{(1)}(k)&\theta ^{-1}\\\theta ^{-1}&k\theta ^{-2}\end{pmatrix}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>I</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
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|
|||
|
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<mo>(</mo>
|
|||
|
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<mtable rowspacing="4pt" columnspacing="1em">
|
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<mtr>
|
|||
|
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<mtd>
|
|||
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<msup>
|
|||
|
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<mi>ψ<!-- ψ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
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|
|||
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|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mtd>
|
|||
|
|
<mtd>
|
|||
|
|
<msup>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mtd>
|
|||
|
|
</mtr>
|
|||
|
|
<mtr>
|
|||
|
|
<mtd>
|
|||
|
|
<msup>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mtd>
|
|||
|
|
<mtd>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<msup>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mtd>
|
|||
|
|
</mtr>
|
|||
|
|
</mtable>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle I(k,\theta )={\begin{pmatrix}\psi ^{(1)}(k)&\theta ^{-1}\\\theta ^{-1}&k\theta ^{-2}\end{pmatrix}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96ca1fa0ecbcf09ce7fc348b6c3363faac0408e4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.412ex; height:6.509ex;" alt="{\displaystyle I(k,\theta )={\begin{pmatrix}\psi ^{(1)}(k)&\theta ^{-1}\\\theta ^{-1}&k\theta ^{-2}\end{pmatrix}}}"></span></td><td class="infobox-data infobox-data-b">
|
|||
|
|
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(\alpha ,\beta )={\begin{pmatrix}\psi ^{(1)}(\alpha )&-\beta ^{-1}\\-\beta ^{-1}&\alpha \beta ^{-2}\end{pmatrix}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>I</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<mtable rowspacing="4pt" columnspacing="1em">
|
|||
|
|
<mtr>
|
|||
|
|
<mtd>
|
|||
|
|
<msup>
|
|||
|
|
<mi>ψ<!-- ψ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mtd>
|
|||
|
|
<mtd>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mtd>
|
|||
|
|
</mtr>
|
|||
|
|
<mtr>
|
|||
|
|
<mtd>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
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</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mtd>
|
|||
|
|
<mtd>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<msup>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mtd>
|
|||
|
|
</mtr>
|
|||
|
|
</mtable>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle I(\alpha ,\beta )={\begin{pmatrix}\psi ^{(1)}(\alpha )&-\beta ^{-1}\\-\beta ^{-1}&\alpha \beta ^{-2}\end{pmatrix}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6881c4895c0e9bbc951e99f996f94e2a3690aae8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.05ex; height:6.509ex;" alt="{\displaystyle I(\alpha ,\beta )={\begin{pmatrix}\psi ^{(1)}(\alpha )&-\beta ^{-1}\\-\beta ^{-1}&\alpha \beta ^{-2}\end{pmatrix}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of Moments</a></th><td class="infobox-data infobox-data-a">
|
|||
|
|
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={\frac {E[X]^{2}}{V[X]}}\quad \quad }">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>E</mi>
|
|||
|
|
<mo stretchy="false">[</mo>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<msup>
|
|||
|
|
<mo stretchy="false">]</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>V</mi>
|
|||
|
|
<mo stretchy="false">[</mo>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mo stretchy="false">]</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mspace width="1em" />
|
|||
|
|
<mspace width="1em" />
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k={\frac {E[X]^{2}}{V[X]}}\quad \quad }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79060985aa8683bbf0b380d57ca56522822342ca" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.895ex; height:6.676ex;" alt="{\displaystyle k={\frac {E[X]^{2}}{V[X]}}\quad \quad }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ={\frac {V[X]}{E[X]}}\quad \quad }">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>V</mi>
|
|||
|
|
<mo stretchy="false">[</mo>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mo stretchy="false">]</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>E</mi>
|
|||
|
|
<mo stretchy="false">[</mo>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mo stretchy="false">]</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mspace width="1em" />
|
|||
|
|
<mspace width="1em" />
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \theta ={\frac {V[X]}{E[X]}}\quad \quad }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de7bf8b64325f4129e05929e2385f3ca37bb88bf" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.731ex; height:6.509ex;" alt="{\displaystyle \theta ={\frac {V[X]}{E[X]}}\quad \quad }"></span></td><td class="infobox-data infobox-data-b">
|
|||
|
|
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ={\frac {E[X]^{2}}{V[X]}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>E</mi>
|
|||
|
|
<mo stretchy="false">[</mo>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<msup>
|
|||
|
|
<mo stretchy="false">]</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>V</mi>
|
|||
|
|
<mo stretchy="false">[</mo>
|
|||
|
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<mi>X</mi>
|
|||
|
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|
|||
|
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|
|||
|
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|
|||
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|
|||
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|
|||
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|
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<annotation encoding="application/x-tex">{\displaystyle \alpha ={\frac {E[X]^{2}}{V[X]}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87074b8ec525badd064920b64dcff7be1c51ceaa" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.526ex; height:6.676ex;" alt="{\displaystyle \alpha ={\frac {E[X]^{2}}{V[X]}}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ={\frac {E[X]}{V[X]}}}">
|
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<semantics>
|
|||
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<mrow class="MJX-TeXAtom-ORD">
|
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<mstyle displaystyle="true" scriptlevel="0">
|
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<mi>β<!-- β --></mi>
|
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<mo>=</mo>
|
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|
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|
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|
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|
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|
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|
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|
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|
|||
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<mi>V</mi>
|
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<mo stretchy="false">[</mo>
|
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<mi>X</mi>
|
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<mo stretchy="false">]</mo>
|
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|
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|
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|
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|
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|
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<annotation encoding="application/x-tex">{\displaystyle \beta ={\frac {E[X]}{V[X]}}}</annotation>
|
|||
|
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</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/187bc571898043026331662ae41bb70d4104d429" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:10.328ex; height:6.509ex;" alt="{\displaystyle \beta ={\frac {E[X]}{V[X]}}}"></span></td></tr></tbody></table>
|
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|
|
<p>In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a>, the <b>gamma distribution</b> is a two-<a href="/wiki/Statistical_parameter" title="Statistical parameter">parameter</a> family of continuous <a href="/wiki/Probability_distribution" title="Probability distribution">probability distributions</a>. The <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a>, <a href="/wiki/Erlang_distribution" title="Erlang distribution">Erlang distribution</a>, and <a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">chi-squared distribution</a> are special cases of the gamma distribution. There are two equivalent parameterizations in common use:
|
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|
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</p>
|
|||
|
|
<ol><li>With a <a href="/wiki/Shape_parameter" title="Shape parameter">shape parameter</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
|
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|
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<mrow class="MJX-TeXAtom-ORD">
|
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|
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<mi>k</mi>
|
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</mstyle>
|
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|
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|
|
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
|
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|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="k"></span> and a <a href="/wiki/Scale_parameter" title="Scale parameter">scale parameter</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">
|
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<semantics>
|
|||
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<mrow class="MJX-TeXAtom-ORD">
|
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<mstyle displaystyle="true" scriptlevel="0">
|
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<mi>θ<!-- θ --></mi>
|
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|
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</mstyle>
|
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</mrow>
|
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|
|
<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>
|
|||
|
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</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="\theta "></span>.</li>
|
|||
|
|
<li>With a shape parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =k}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \alpha =k}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/492bbf1283920d57a534b63fca94a2e2ff4d1946" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.797ex; height:2.176ex;" alt="{\displaystyle \alpha =k}"></span> and an inverse scale parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =1/\theta }">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>/</mo>
|
|||
|
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</mrow>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \beta =1/\theta }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/664250d7209fe5e66b2de5a0c74c7a28ef17b395" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.846ex; height:2.843ex;" alt="{\displaystyle \beta =1/\theta }"></span> , called a <a href="/wiki/Rate_parameter" class="mw-redirect" title="Rate parameter">rate parameter</a>.</li></ol>
|
|||
|
|
<p>In each of these forms, both parameters are positive real numbers.
|
|||
|
|
</p><p>The gamma distribution is the <a href="/wiki/Maximum_entropy_probability_distribution" title="Maximum entropy probability distribution">maximum entropy probability distribution</a> (both with respect to a uniform base measure and a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/x}">
|
|||
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<semantics>
|
|||
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|
<mrow class="MJX-TeXAtom-ORD">
|
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<mstyle displaystyle="true" scriptlevel="0">
|
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|
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<mn>1</mn>
|
|||
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<mrow class="MJX-TeXAtom-ORD">
|
|||
|
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<mo>/</mo>
|
|||
|
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|
|||
|
|
<mi>x</mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle 1/x}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a55fefc6f37f48a9b4414b09ad3b17dfa739d9e3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="1/x"></span> base measure) for a random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>X</mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> for which <b>E</b>[<i>X</i>] = <i>kθ</i> = <i>α</i>/<i>β</i> is fixed and greater than zero, and <b>E</b>[ln(<i>X</i>)] = <i>ψ</i>(<i>k</i>) + ln(<i>θ</i>) = <i>ψ</i>(<i>α</i>) − ln(<i>β</i>) is fixed (<i>ψ</i> is the <a href="/wiki/Digamma_function" title="Digamma function">digamma function</a>).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[1]</a></sup>
|
|||
|
|
</p>
|
|||
|
|
<meta property="mw:PageProp/toc" />
|
|||
|
|
<h2><span class="mw-headline" id="Definitions">Definitions</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=1" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
|
|||
|
|
<p>The parameterization with <i>k</i> and <i>θ</i> appears to be more common in <a href="/wiki/Econometrics" title="Econometrics">econometrics</a> and other applied fields, where the gamma distribution is frequently used to model waiting times. For instance, in <a href="/wiki/Accelerated_life_testing" title="Accelerated life testing">life testing</a>, the waiting time until death is a <a href="/wiki/Random_variable" title="Random variable">random variable</a> that is frequently modeled with a gamma distribution. See Hogg and Craig<sup id="cite_ref-2" class="reference"><a href="#cite_note-2">[2]</a></sup> for an explicit motivation.
|
|||
|
|
</p><p>The parameterization with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="\alpha "></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="\beta "></span> is more common in <a href="/wiki/Bayesian_statistics" title="Bayesian statistics">Bayesian statistics</a>, where the gamma distribution is used as a <a href="/wiki/Conjugate_prior" title="Conjugate prior">conjugate prior</a> distribution for various types of inverse scale (rate) parameters, such as the <i>λ</i> of an <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a> or a <a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distribution</a><sup id="cite_ref-3" class="reference"><a href="#cite_note-3">[3]</a></sup> – or for that matter, the <i>β</i> of the gamma distribution itself. The closely related <a href="/wiki/Inverse-gamma_distribution" title="Inverse-gamma distribution">inverse-gamma distribution</a> is used as a conjugate prior for scale parameters, such as the <a href="/wiki/Variance" title="Variance">variance</a> of a <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>.
|
|||
|
|
</p><p>If <i>k</i> is a positive <a href="/wiki/Integer" title="Integer">integer</a>, then the distribution represents an <a href="/wiki/Erlang_distribution" title="Erlang distribution">Erlang distribution</a>; i.e., the sum of <i>k</i> independent <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponentially distributed</a> <a href="/wiki/Random_variable" title="Random variable">random variables</a>, each of which has a mean of <i>θ</i>.
|
|||
|
|
</p>
|
|||
|
|
<h3><span id="Characterization_using_shape_.CE.B1_and_rate_.CE.B2"></span><span class="mw-headline" id="Characterization_using_shape_α_and_rate_β">Characterization using shape <i>α</i> and rate <i>β</i></span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=2" title="Edit section: Characterization using shape α and rate β"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
|
|||
|
|
<p>The gamma distribution can be parameterized in terms of a <a href="/wiki/Shape_parameter" title="Shape parameter">shape parameter</a> <i>α</i> = <i>k</i> and an inverse scale parameter <i>β</i> = 1/<i>θ</i>, called a <a href="/wiki/Rate_parameter" class="mw-redirect" title="Rate parameter">rate parameter</a>. A random variable <i>X</i> that is gamma-distributed with shape <i>α</i> and rate <i>β</i> is denoted
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \Gamma (\alpha ,\beta )\equiv \operatorname {Gamma} (\alpha ,\beta )}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
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<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mo>∼<!-- ∼ --></mo>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>≡<!-- ≡ --></mo>
|
|||
|
|
<mi>Gamma</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle X\sim \Gamma (\alpha ,\beta )\equiv \operatorname {Gamma} (\alpha ,\beta )}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57e9787d3e7e96c76c9b0e247fecbf8d3b25335d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.976ex; height:2.843ex;" alt="{\displaystyle X\sim \Gamma (\alpha ,\beta )\equiv \operatorname {Gamma} (\alpha ,\beta )}"></span></dd></dl>
|
|||
|
|
<p>The corresponding probability density function in the shape-rate parameterization is
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}f(x;\alpha ,\beta )&={\frac {x^{\alpha -1}e^{-\beta x}\beta ^{\alpha }}{\Gamma (\alpha )}}\quad {\text{ for }}x>0\quad \alpha ,\beta >0,\\[6pt]\end{aligned}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
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|
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<mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
|
|||
|
|
<mtr>
|
|||
|
|
<mtd>
|
|||
|
|
<mi>f</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo>;</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mtd>
|
|||
|
|
<mtd>
|
|||
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|
|||
|
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<mo>=</mo>
|
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|
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<mrow class="MJX-TeXAtom-ORD">
|
|||
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<mfrac>
|
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|
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|
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|
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|
|||
|
|
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|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
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<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
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|
|||
|
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</msup>
|
|||
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<msup>
|
|||
|
|
<mi>e</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<msup>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mspace width="1em" />
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mtext> for </mtext>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo>></mo>
|
|||
|
|
<mn>0</mn>
|
|||
|
|
<mspace width="1em" />
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mo>></mo>
|
|||
|
|
<mn>0</mn>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
</mtd>
|
|||
|
|
</mtr>
|
|||
|
|
</mtable>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f(x;\alpha ,\beta )&={\frac {x^{\alpha -1}e^{-\beta x}\beta ^{\alpha }}{\Gamma (\alpha )}}\quad {\text{ for }}x>0\quad \alpha ,\beta >0,\\[6pt]\end{aligned}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf760a328d5b468fea5f9f1d47cca54b558b6da" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:48.747ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}f(x;\alpha ,\beta )&={\frac {x^{\alpha -1}e^{-\beta x}\beta ^{\alpha }}{\Gamma (\alpha )}}\quad {\text{ for }}x>0\quad \alpha ,\beta >0,\\[6pt]\end{aligned}}}"></span></dd></dl>
|
|||
|
|
<p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (\alpha )}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \Gamma (\alpha )}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffad1626eab58fe191891edd57e7d27fa9a2def8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.75ex; height:2.843ex;" alt="{\displaystyle \Gamma (\alpha )}"></span> is the <a href="/wiki/Gamma_function" title="Gamma function">gamma function</a>.
|
|||
|
|
For all positive integers, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (\alpha )=(\alpha -1)!}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>!</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \Gamma (\alpha )=(\alpha -1)!}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df21f16281b82ecb7a2dc5132fc54e88741b6855" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.795ex; height:2.843ex;" alt="{\displaystyle \Gamma (\alpha )=(\alpha -1)!}"></span>.
|
|||
|
|
</p><p>The <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> is the regularized gamma function:
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x;\alpha ,\beta )=\int _{0}^{x}f(u;\alpha ,\beta )\,du={\frac {\gamma (\alpha ,\beta x)}{\Gamma (\alpha )}},}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>F</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo>;</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<msubsup>
|
|||
|
|
<mo>∫<!-- ∫ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>0</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>x</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msubsup>
|
|||
|
|
<mi>f</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>u</mi>
|
|||
|
|
<mo>;</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mspace width="thinmathspace" />
|
|||
|
|
<mi>d</mi>
|
|||
|
|
<mi>u</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>γ<!-- γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle F(x;\alpha ,\beta )=\int _{0}^{x}f(u;\alpha ,\beta )\,du={\frac {\gamma (\alpha ,\beta x)}{\Gamma (\alpha )}},}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad6e29c7cfefa404f14fea26c52ea1471116570c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.963ex; height:6.509ex;" alt="{\displaystyle F(x;\alpha ,\beta )=\int _{0}^{x}f(u;\alpha ,\beta )\,du={\frac {\gamma (\alpha ,\beta x)}{\Gamma (\alpha )}},}"></span></dd></dl>
|
|||
|
|
<p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (\alpha ,\beta x)}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>γ<!-- γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \gamma (\alpha ,\beta x)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0387febf90a45b820b486bf53b5a89a6d14fa664" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.255ex; height:2.843ex;" alt="{\displaystyle \gamma (\alpha ,\beta x)}"></span> is the lower <a href="/wiki/Incomplete_gamma_function" title="Incomplete gamma function">incomplete gamma function</a>.
|
|||
|
|
</p><p>If <i>α</i> is a positive <a href="/wiki/Integer" title="Integer">integer</a> (i.e., the distribution is an <a href="/wiki/Erlang_distribution" title="Erlang distribution">Erlang distribution</a>), the cumulative distribution function has the following series expansion:<sup id="cite_ref-Papoulis_4-0" class="reference"><a href="#cite_note-Papoulis-4">[4]</a></sup>
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x;\alpha ,\beta )=1-\sum _{i=0}^{\alpha -1}{\frac {(\beta x)^{i}}{i!}}e^{-\beta x}=e^{-\beta x}\sum _{i=\alpha }^{\infty }{\frac {(\beta x)^{i}}{i!}}.}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>F</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo>;</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<munderover>
|
|||
|
|
<mo>∑<!-- ∑ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>0</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</munderover>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mrow>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<msup>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>!</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<msup>
|
|||
|
|
<mi>e</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>e</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<munderover>
|
|||
|
|
<mo>∑<!-- ∑ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi mathvariant="normal">∞<!-- ∞ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</munderover>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mrow>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<msup>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>!</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>.</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle F(x;\alpha ,\beta )=1-\sum _{i=0}^{\alpha -1}{\frac {(\beta x)^{i}}{i!}}e^{-\beta x}=e^{-\beta x}\sum _{i=\alpha }^{\infty }{\frac {(\beta x)^{i}}{i!}}.}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c56150efa42bf73a899c72690ba1004ffcfa397e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:49.652ex; height:7.343ex;" alt="{\displaystyle F(x;\alpha ,\beta )=1-\sum _{i=0}^{\alpha -1}{\frac {(\beta x)^{i}}{i!}}e^{-\beta x}=e^{-\beta x}\sum _{i=\alpha }^{\infty }{\frac {(\beta x)^{i}}{i!}}.}"></span></dd></dl>
|
|||
|
|
<h3><span id="Characterization_using_shape_k_and_scale_.CE.B8"></span><span class="mw-headline" id="Characterization_using_shape_k_and_scale_θ">Characterization using shape <i>k</i> and scale <i>θ</i></span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=3" title="Edit section: Characterization using shape k and scale θ"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
|
|||
|
|
<p>A random variable <i>X</i> that is gamma-distributed with shape <i>k</i> and scale <i>θ</i> is denoted by
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \Gamma (k,\theta )\equiv \operatorname {Gamma} (k,\theta )}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mo>∼<!-- ∼ --></mo>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>≡<!-- ≡ --></mo>
|
|||
|
|
<mi>Gamma</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle X\sim \Gamma (k,\theta )\equiv \operatorname {Gamma} (k,\theta )}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b0320535d26cf0c3b96e123ab0a046b6db3b1f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.941ex; height:2.843ex;" alt="{\displaystyle X\sim \Gamma (k,\theta )\equiv \operatorname {Gamma} (k,\theta )}"></span></dd></dl>
|
|||
|
|
<figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Gamma-PDF-3D.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Gamma-PDF-3D.png/320px-Gamma-PDF-3D.png" decoding="async" width="320" height="259" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Gamma-PDF-3D.png/480px-Gamma-PDF-3D.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Gamma-PDF-3D.png/640px-Gamma-PDF-3D.png 2x" data-file-width="2040" data-file-height="1653" /></a><figcaption>Illustration of the gamma PDF for parameter values over <i>k</i> and <i>x</i> with <i>θ</i> set to 1, 2, 3, 4, 5 and 6. One can see each <i>θ</i> layer by itself here <a class="external autonumber" href="https://commons.wikimedia.org/wiki/File:Gamma-PDF-3D-by-k.png">[2]</a> as well as by <i>k</i> <a class="external autonumber" href="https://commons.wikimedia.org/wiki/File:Gamma-PDF-3D-by-Theta.png">[3]</a> and <i>x</i>. <a class="external autonumber" href="https://commons.wikimedia.org/wiki/File:Gamma-PDF-3D-by-x.png">[4]</a>.</figcaption></figure>
|
|||
|
|
<p>The <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> using the shape-scale parametrization is
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x;k,\theta )={\frac {x^{k-1}e^{-x/\theta }}{\theta ^{k}\Gamma (k)}}\quad {\text{ for }}x>0{\text{ and }}k,\theta >0.}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>f</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo>;</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mrow>
|
|||
|
|
<msup>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<msup>
|
|||
|
|
<mi>e</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>/</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<msup>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mspace width="1em" />
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mtext> for </mtext>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo>></mo>
|
|||
|
|
<mn>0</mn>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mtext> and </mtext>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo>></mo>
|
|||
|
|
<mn>0.</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle f(x;k,\theta )={\frac {x^{k-1}e^{-x/\theta }}{\theta ^{k}\Gamma (k)}}\quad {\text{ for }}x>0{\text{ and }}k,\theta >0.}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caf176962d326ad7af8186d5f4cd3f3e7fae4852" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:47.381ex; height:6.509ex;" alt="{\displaystyle f(x;k,\theta )={\frac {x^{k-1}e^{-x/\theta }}{\theta ^{k}\Gamma (k)}}\quad {\text{ for }}x>0{\text{ and }}k,\theta >0.}"></span></dd></dl>
|
|||
|
|
<p>Here Γ(<i>k</i>) is the <a href="/wiki/Gamma_function" title="Gamma function">gamma function</a> evaluated at <i>k</i>.
|
|||
|
|
</p><p>The <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> is the regularized gamma function:
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x;k,\theta )=\int _{0}^{x}f(u;k,\theta )\,du={\frac {\gamma \left(k,{\frac {x}{\theta }}\right)}{\Gamma (k)}},}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>F</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo>;</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<msubsup>
|
|||
|
|
<mo>∫<!-- ∫ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>0</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>x</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msubsup>
|
|||
|
|
<mi>f</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>u</mi>
|
|||
|
|
<mo>;</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mspace width="thinmathspace" />
|
|||
|
|
<mi>d</mi>
|
|||
|
|
<mi>u</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>γ<!-- γ --></mi>
|
|||
|
|
<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle F(x;k,\theta )=\int _{0}^{x}f(u;k,\theta )\,du={\frac {\gamma \left(k,{\frac {x}{\theta }}\right)}{\Gamma (k)}},}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e10c41c86ac78205b3f7cd5a1ba84052714ff662" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.119ex; height:8.343ex;" alt="{\displaystyle F(x;k,\theta )=\int _{0}^{x}f(u;k,\theta )\,du={\frac {\gamma \left(k,{\frac {x}{\theta }}\right)}{\Gamma (k)}},}"></span></dd></dl>
|
|||
|
|
<p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \left(k,{\frac {x}{\theta }}\right)}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>γ<!-- γ --></mi>
|
|||
|
|
<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \gamma \left(k,{\frac {x}{\theta }}\right)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bab1607e067f732c5fc272613cb87e793c79af0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.836ex; height:5.009ex;" alt="{\displaystyle \gamma \left(k,{\frac {x}{\theta }}\right)}"></span> is the lower <a href="/wiki/Incomplete_gamma_function" title="Incomplete gamma function">incomplete gamma function</a>.
|
|||
|
|
</p><p>It can also be expressed as follows, if <i>k</i> is a positive <a href="/wiki/Integer" title="Integer">integer</a> (i.e., the distribution is an <a href="/wiki/Erlang_distribution" title="Erlang distribution">Erlang distribution</a>):<sup id="cite_ref-Papoulis_4-1" class="reference"><a href="#cite_note-Papoulis-4">[4]</a></sup>
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x;k,\theta )=1-\sum _{i=0}^{k-1}{\frac {1}{i!}}\left({\frac {x}{\theta }}\right)^{i}e^{-x/\theta }=e^{-x/\theta }\sum _{i=k}^{\infty }{\frac {1}{i!}}\left({\frac {x}{\theta }}\right)^{i}.}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>F</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo>;</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<munderover>
|
|||
|
|
<mo>∑<!-- ∑ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>0</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</munderover>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>!</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<msup>
|
|||
|
|
<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<msup>
|
|||
|
|
<mi>e</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>/</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>e</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>/</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<munderover>
|
|||
|
|
<mo>∑<!-- ∑ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi mathvariant="normal">∞<!-- ∞ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</munderover>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>!</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<msup>
|
|||
|
|
<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo>.</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle F(x;k,\theta )=1-\sum _{i=0}^{k-1}{\frac {1}{i!}}\left({\frac {x}{\theta }}\right)^{i}e^{-x/\theta }=e^{-x/\theta }\sum _{i=k}^{\infty }{\frac {1}{i!}}\left({\frac {x}{\theta }}\right)^{i}.}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9b50954cd5d3328b10d9922b97b893c25cd76a8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:54.275ex; height:7.343ex;" alt="{\displaystyle F(x;k,\theta )=1-\sum _{i=0}^{k-1}{\frac {1}{i!}}\left({\frac {x}{\theta }}\right)^{i}e^{-x/\theta }=e^{-x/\theta }\sum _{i=k}^{\infty }{\frac {1}{i!}}\left({\frac {x}{\theta }}\right)^{i}.}"></span></dd></dl>
|
|||
|
|
<p>Both parametrizations are common because either can be more convenient depending on the situation.
|
|||
|
|
</p>
|
|||
|
|
<h2><span class="mw-headline" id="Properties">Properties</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=4" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
|
|||
|
|
<h3><span class="mw-headline" id="Mean_and_variance">Mean and variance</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=5" title="Edit section: Mean and variance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
|
|||
|
|
<p>The mean of gamma distribution is given by the product of its shape and scale parameters:
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =k\theta =\alpha /\beta }">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>μ<!-- μ --></mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>/</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \mu =k\theta =\alpha /\beta }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6a58ddcce52712f203ad4142baba4a474b8bef8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.882ex; height:2.843ex;" alt="{\displaystyle \mu =k\theta =\alpha /\beta }"></span></dd></dl>
|
|||
|
|
<p>The variance is:
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}=k\theta ^{2}=\alpha /\beta ^{2}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<msup>
|
|||
|
|
<mi>σ<!-- σ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<msup>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>/</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<msup>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}=k\theta ^{2}=\alpha /\beta ^{2}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c88cc5e24c9d0eb7176ce48ec7069281aea69a2d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.979ex; height:3.176ex;" alt="{\displaystyle \sigma ^{2}=k\theta ^{2}=\alpha /\beta ^{2}}"></span></dd></dl>
|
|||
|
|
<p>The square root of the inverse shape parameter gives the <a href="/wiki/Coefficient_of_variation" title="Coefficient of variation">coefficient of variation</a>:
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma /\mu =k^{-0.5}=1/{\sqrt {\alpha }}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>σ<!-- σ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>/</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>μ<!-- μ --></mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>0.5</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>/</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<msqrt>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
</msqrt>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \sigma /\mu =k^{-0.5}=1/{\sqrt {\alpha }}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52db1d7831315dcadbff298e415c9fc48de55076" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.663ex; height:3.343ex;" alt="{\displaystyle \sigma /\mu =k^{-0.5}=1/{\sqrt {\alpha }}}"></span></dd></dl>
|
|||
|
|
<h3><span class="mw-headline" id="Skewness">Skewness</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=6" title="Edit section: Skewness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
|
|||
|
|
<p>The <a href="/wiki/Skewness" title="Skewness">skewness</a> of the gamma distribution only depends on its shape parameter, <i>k</i>, and it is equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2/{\sqrt {k}}.}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>/</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<msqrt>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</msqrt>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>.</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle 2/{\sqrt {k}}.}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7958ea261b36951e15f1c9363cd01a384a8217d9" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.119ex; height:3.176ex;" alt="{\displaystyle 2/{\sqrt {k}}.}"></span>
|
|||
|
|
</p>
|
|||
|
|
<h3><span class="mw-headline" id="Higher_moments">Higher moments</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=7" title="Edit section: Higher moments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
|
|||
|
|
<p>The <i>n</i>th <a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">raw moment</a> is given by:
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {E} [X^{n}]=\theta ^{n}{\frac {\Gamma (k+n)}{\Gamma (k)}}=\theta ^{n}\prod _{i=1}^{n}(k+i-1)\;{\text{ for }}n=1,2,\ldots .}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi mathvariant="normal">E</mi>
|
|||
|
|
</mrow>
|
|||
|
|
<mo stretchy="false">[</mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>n</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo stretchy="false">]</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>n</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mrow>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mi>n</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>n</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<munderover>
|
|||
|
|
<mo>∏<!-- ∏ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>n</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</munderover>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mspace width="thickmathspace" />
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mtext> for </mtext>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>n</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mo>…<!-- … --></mo>
|
|||
|
|
<mo>.</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \mathrm {E} [X^{n}]=\theta ^{n}{\frac {\Gamma (k+n)}{\Gamma (k)}}=\theta ^{n}\prod _{i=1}^{n}(k+i-1)\;{\text{ for }}n=1,2,\ldots .}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67b2480d1a1107cd132f94b5280db23ddde59033" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:57.709ex; height:6.843ex;" alt="{\displaystyle \mathrm {E} [X^{n}]=\theta ^{n}{\frac {\Gamma (k+n)}{\Gamma (k)}}=\theta ^{n}\prod _{i=1}^{n}(k+i-1)\;{\text{ for }}n=1,2,\ldots .}"></span></dd></dl>
|
|||
|
|
<h3><span class="mw-headline" id="Median_approximations_and_bounds">Median approximations and bounds</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=8" title="Edit section: Median approximations and bounds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
|
|||
|
|
<figure typeof="mw:File/Thumb"><a href="/wiki/File:Gamma_distribution_median_bounds.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Gamma_distribution_median_bounds.png/320px-Gamma_distribution_median_bounds.png" decoding="async" width="320" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Gamma_distribution_median_bounds.png/480px-Gamma_distribution_median_bounds.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/42/Gamma_distribution_median_bounds.png/640px-Gamma_distribution_median_bounds.png 2x" data-file-width="2333" data-file-height="1750" /></a><figcaption>Bounds and asymptotic approximations to the median of the gamma distribution. The cyan-colored region indicates the large gap between published lower and upper bounds.</figcaption></figure>
|
|||
|
|
<p>Unlike the mode and the mean, which have readily calculable formulas based on the parameters, the median does not have a closed-form equation. The median for this distribution is the value <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="\nu "></span> such that
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\Gamma (k)\theta ^{k}}}\int _{0}^{\nu }x^{k-1}e^{-x/\theta }dx={\frac {1}{2}}.}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mrow>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<msubsup>
|
|||
|
|
<mo>∫<!-- ∫ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>0</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msubsup>
|
|||
|
|
<msup>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<msup>
|
|||
|
|
<mi>e</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>/</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mi>d</mi>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>.</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\Gamma (k)\theta ^{k}}}\int _{0}^{\nu }x^{k-1}e^{-x/\theta }dx={\frac {1}{2}}.}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/932c212790dbeb09a8b9244e3eeb9b8fbb31b309" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.766ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\Gamma (k)\theta ^{k}}}\int _{0}^{\nu }x^{k-1}e^{-x/\theta }dx={\frac {1}{2}}.}"></span></dd></dl>
|
|||
|
|
<p>A rigorous treatment of the problem of determining an asymptotic expansion and bounds for the median of the gamma distribution was handled first by Chen and Rubin, who proved that (for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =1}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \theta =1}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f287a8873fea8959af7d41cd891678d8875d3a15" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle \theta =1}"></span>)
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k-{\frac {1}{3}}<\nu (k)<k,}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mn>3</mn>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo><</mo>
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo><</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k-{\frac {1}{3}}<\nu (k)<k,}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50148b370f2224a88b71230d9b89d393ce2a35ae" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.358ex; height:5.176ex;" alt="{\displaystyle k-{\frac {1}{3}}<\nu (k)<k,}"></span></dd></dl>
|
|||
|
|
<p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (k)=k}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>μ<!-- μ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \mu (k)=k}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7dd26b6f7f7f39a301f14b3c05dff07882ba906" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.732ex; height:2.843ex;" alt="{\displaystyle \mu (k)=k}"></span> is the mean and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (k)}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \nu (k)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7023bd901ef6b557d850ffa2089200e625d370e7" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.253ex; height:2.843ex;" alt="{\displaystyle \nu (k)}"></span> is the median of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Gamma}}(k,1)}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mtext>Gamma</mtext>
|
|||
|
|
</mrow>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\text{Gamma}}(k,1)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e0648b913658b37bf6c4db6034c038b8e385e2d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.238ex; height:2.843ex;" alt="{\displaystyle {\text{Gamma}}(k,1)}"></span> distribution.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5">[5]</a></sup> For other values of the scale parameter, the mean scales to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =k\theta }">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>μ<!-- μ --></mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \mu =k\theta }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2e18aba75babe49bc0b1eb2a0e35726e395b93b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.802ex; height:2.676ex;" alt="{\displaystyle \mu =k\theta }"></span>, and the median bounds and approximations would be similarly scaled by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="\theta "></span>.
|
|||
|
|
</p><p>K. P. Choi found the first five terms in a <a href="/wiki/Laurent_series" title="Laurent series">Laurent series</a> asymptotic approximation of the median by comparing the median to <a href="/wiki/Ramanujan_theta_function" title="Ramanujan theta function">Ramanujan's <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="\theta "></span> function</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6">[6]</a></sup> Berg and Pedersen found more terms:<sup id="cite_ref-berg_7-0" class="reference"><a href="#cite_note-berg-7">[7]</a></sup>
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (k)=k-{\frac {1}{3}}+{\frac {8}{405k}}+{\frac {184}{25515k^{2}}}+{\frac {2248}{3444525k^{3}}}-{\frac {19006408}{15345358875k^{4}}}-O\left({\frac {1}{k^{5}}}\right)+\cdots }">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mn>3</mn>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>8</mn>
|
|||
|
|
<mrow>
|
|||
|
|
<mn>405</mn>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>184</mn>
|
|||
|
|
<mrow>
|
|||
|
|
<mn>25515</mn>
|
|||
|
|
<msup>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>2248</mn>
|
|||
|
|
<mrow>
|
|||
|
|
<mn>3444525</mn>
|
|||
|
|
<msup>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>3</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>19006408</mn>
|
|||
|
|
<mrow>
|
|||
|
|
<mn>15345358875</mn>
|
|||
|
|
<msup>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>4</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>O</mi>
|
|||
|
|
<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<msup>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>5</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mo>⋯<!-- ⋯ --></mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \nu (k)=k-{\frac {1}{3}}+{\frac {8}{405k}}+{\frac {184}{25515k^{2}}}+{\frac {2248}{3444525k^{3}}}-{\frac {19006408}{15345358875k^{4}}}-O\left({\frac {1}{k^{5}}}\right)+\cdots }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06425d870fa9b7a504eaaef775e51f7937ac4b24" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:83.426ex; height:6.176ex;" alt="{\displaystyle \nu (k)=k-{\frac {1}{3}}+{\frac {8}{405k}}+{\frac {184}{25515k^{2}}}+{\frac {2248}{3444525k^{3}}}-{\frac {19006408}{15345358875k^{4}}}-O\left({\frac {1}{k^{5}}}\right)+\cdots }"></span></dd></dl>
|
|||
|
|
<figure typeof="mw:File/Thumb"><a href="/wiki/File:Gamma_distribution_median_Lyon_bounds.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Gamma_distribution_median_Lyon_bounds.png/320px-Gamma_distribution_median_Lyon_bounds.png" decoding="async" width="320" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Gamma_distribution_median_Lyon_bounds.png/480px-Gamma_distribution_median_Lyon_bounds.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Gamma_distribution_median_Lyon_bounds.png/640px-Gamma_distribution_median_Lyon_bounds.png 2x" data-file-width="2333" data-file-height="1750" /></a><figcaption>Two gamma distribution median asymptotes which were proved in 2023 to be bounds (upper solid red and lower dashed red), of the from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (k)\approx 2^{-1/k}(A+k)}">
|
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|
|||
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|
|||
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|
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|
|||
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|
|||
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|
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|
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|
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|
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|
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|
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|
|||
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|
|||
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|
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|
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|
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|
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|
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|
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<annotation encoding="application/x-tex">{\displaystyle \nu (k)\approx 2^{-1/k}(A+k)}</annotation>
|
|||
|
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</semantics>
|
|||
|
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de282e9589256ce4d8b25101b4a64b840fa4c820" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.129ex; height:3.343ex;" alt="{\displaystyle \nu (k)\approx 2^{-1/k}(A+k)}"></span>, and an interpolation between them that makes an approximation (dotted red) that is exact at k = 1 and has maximum relative error of about 0.6%. The cyan shaded region is the remaining gap between upper and lower bounds (or conjectured bounds), including these new bounds and the bounds in the previous figure.</figcaption></figure>
|
|||
|
|
<figure typeof="mw:File/Thumb"><a href="/wiki/File:Gamma_distribution_median_loglog_bounds.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Gamma_distribution_median_loglog_bounds.png/320px-Gamma_distribution_median_loglog_bounds.png" decoding="async" width="320" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Gamma_distribution_median_loglog_bounds.png/480px-Gamma_distribution_median_loglog_bounds.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Gamma_distribution_median_loglog_bounds.png/640px-Gamma_distribution_median_loglog_bounds.png 2x" data-file-width="2333" data-file-height="1750" /></a><figcaption><a href="/wiki/Log%E2%80%93log_plot" title="Log–log plot">Log–log plot</a> of upper (solid) and lower (dashed) bounds to the median of a gamma distribution and the gaps between them. The green, yellow, and cyan regions represent the gap before the Lyon 2021 paper. The green and yellow narrow that gap with the lower bounds that Lyon proved. Lyon's conjectured bounds further narrow the yellow. Mostly within the yellow, closed-form rational-function-interpolated bounds are plotted along with the numerically calculated median (dotted) value. Tighter interpolated bounds exist but are not plotted, as they would not be resolved at this scale.</figcaption></figure>
|
|||
|
|
<p>Partial sums of these series are good approximations for high enough <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
|
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|
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|
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|
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|
|||
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|
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|
|||
|
|
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|
|||
|
|
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|
|||
|
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|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="k"></span>; they are not plotted in the figure, which is focused on the low-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
|
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|
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|
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|
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|
|||
|
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|
|||
|
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|
|||
|
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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="k"></span> region that is less well approximated.
|
|||
|
|
</p><p>Berg and Pedersen also proved many properties of the median, showing that it is a convex function of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
|
|||
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<semantics>
|
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|
|||
|
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|
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|
|||
|
|
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|
|||
|
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|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
|
|||
|
|
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|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="k"></span>,<sup id="cite_ref-convex_8-0" class="reference"><a href="#cite_note-convex-8">[8]</a></sup> and that the asymptotic behavior near <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
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|
|||
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|
|||
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|
<mo>=</mo>
|
|||
|
|
<mn>0</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k=0}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6307c8a99dad7d0bcb712352ae0a748bd99a038b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="k=0"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (k)\approx e^{-\gamma }2^{-1/k}}">
|
|||
|
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<semantics>
|
|||
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|
|||
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|
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|
|||
|
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<mo stretchy="false">(</mo>
|
|||
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|
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|
|||
|
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|
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<msup>
|
|||
|
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<mi>e</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
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<mo>−<!-- − --></mo>
|
|||
|
|
<mi>γ<!-- γ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<msup>
|
|||
|
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<mn>2</mn>
|
|||
|
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<mrow class="MJX-TeXAtom-ORD">
|
|||
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<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
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|
|||
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<mo>/</mo>
|
|||
|
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|
|||
|
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|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \nu (k)\approx e^{-\gamma }2^{-1/k}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f8e0dbcff88172cb178dc51b01046ea6a4a0c9f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.012ex; height:3.343ex;" alt="{\displaystyle \nu (k)\approx e^{-\gamma }2^{-1/k}}"></span> (where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">
|
|||
|
|
<semantics>
|
|||
|
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<mrow class="MJX-TeXAtom-ORD">
|
|||
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<mstyle displaystyle="true" scriptlevel="0">
|
|||
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<mi>γ<!-- γ --></mi>
|
|||
|
|
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|
|||
|
|
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|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="\gamma "></span> is the <a href="/wiki/Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a>), and that for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>0}">
|
|||
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|
|||
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|
|||
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|
|||
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|
|||
|
|
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|
|||
|
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|
|||
|
|
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|
|||
|
|
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|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k>0}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27b3af208b148139eefc03f0f80fa94c38c5af45" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="k>0"></span> the median is bounded by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k2^{-1/k}<\nu (k)<ke^{-1/3k}}">
|
|||
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|
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
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|
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|
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|
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|
|||
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|
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|
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|
|||
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|
|||
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|
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|
|||
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|
<mo><</mo>
|
|||
|
|
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|
|||
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|
|||
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|
|||
|
|
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|
|||
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|
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|
|||
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|
|||
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|
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|
|||
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|
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|
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|
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|
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|
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|
|||
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|
|||
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|
|||
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|
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|
|||
|
|
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|
|||
|
|
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|
|||
|
|
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|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k2^{-1/k}<\nu (k)<ke^{-1/3k}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89f7ba6d1b9eabdd7621fc99f96065e5b1f96dda" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.962ex; height:3.343ex;" alt="{\displaystyle k2^{-1/k}<\nu (k)<ke^{-1/3k}}"></span>.<sup id="cite_ref-berg_7-1" class="reference"><a href="#cite_note-berg-7">[7]</a></sup>
|
|||
|
|
</p><p>A closer linear upper bound, for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 1}">
|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
|
|
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|
|||
|
|
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|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k\geq 1}</annotation>
|
|||
|
|
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|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30d7dcf305b7bce39d36df72fe3985b47aa9961" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.472ex; height:2.343ex;" alt="k \ge 1"></span> only, was provided in 2021 by Gaunt and Merkle,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9">[9]</a></sup> relying on the Berg and Pedersen result that the slope of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (k)}">
|
|||
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|
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|
|||
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|
|||
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|
|||
|
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|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
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|
|||
|
|
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|
|||
|
|
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|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \nu (k)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7023bd901ef6b557d850ffa2089200e625d370e7" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.253ex; height:2.843ex;" alt="{\displaystyle \nu (k)}"></span> is everywhere less than 1:
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (k)\leq k-1+\log 2~~}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
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<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>≤<!-- ≤ --></mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mi>log</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
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|
|||
|
|
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|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \nu (k)\leq k-1+\log 2~~}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92b15d0fc8940dbe55f21e5697550d78722c4983" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.088ex; height:2.843ex;" alt="{\displaystyle \nu (k)\leq k-1+\log 2~~}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 1}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>≥<!-- ≥ --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k\geq 1}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30d7dcf305b7bce39d36df72fe3985b47aa9961" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.472ex; height:2.343ex;" alt="k \ge 1"></span> (with equality at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k=1}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="k=1"></span>)</dd></dl>
|
|||
|
|
<p>which can be extended to a bound for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>0}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>></mo>
|
|||
|
|
<mn>0</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k>0}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27b3af208b148139eefc03f0f80fa94c38c5af45" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="k>0"></span> by taking the max with the chord shown in the figure, since the median was proved convex.<sup id="cite_ref-convex_8-1" class="reference"><a href="#cite_note-convex-8">[8]</a></sup>
|
|||
|
|
</p><p>An approximation to the median that is asymptotically accurate at high <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="k"></span> and reasonable down to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0.5}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>0.5</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k=0.5}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aea83a57db153f846b8b7b1a8306e59c4e43ef0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.281ex; height:2.176ex;" alt="{\displaystyle k=0.5}"></span> or a bit lower follows from the <a href="/wiki/Wilson%E2%80%93Hilferty_transformation" class="mw-redirect" title="Wilson–Hilferty transformation">Wilson–Hilferty transformation</a>:
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (k)=k\left(1-{\frac {1}{9k}}\right)^{3}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<msup>
|
|||
|
|
<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<mrow>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mrow>
|
|||
|
|
<mn>9</mn>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>3</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \nu (k)=k\left(1-{\frac {1}{9k}}\right)^{3}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4061f7fcbab5e475380fda6e2dfcf445d100e0dc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.25ex; height:6.509ex;" alt="{\displaystyle \nu (k)=k\left(1-{\frac {1}{9k}}\right)^{3}}"></span></dd></dl>
|
|||
|
|
<p>which goes negative for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k<1/9}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo><</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>/</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mn>9</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k<1/9}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b68a68b15e862c49b0f09a218ff1a94b061271e8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.797ex; height:2.843ex;" alt="{\displaystyle k<1/9}"></span>.
|
|||
|
|
</p><p>In 2021, Lyon proposed several approximations of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (k)\approx 2^{-1/k}(A+Bk)}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>≈<!-- ≈ --></mo>
|
|||
|
|
<msup>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>/</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>A</mi>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mi>B</mi>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \nu (k)\approx 2^{-1/k}(A+Bk)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f59966fa882b4fdd8a8abbfe3882d7b0e5cbedda" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.893ex; height:3.343ex;" alt="{\displaystyle \nu (k)\approx 2^{-1/k}(A+Bk)}"></span>. He conjectured values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>A</mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>B</mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle B}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="B"></span> for which this approximation is an asymptotically tight upper or lower bound for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>0}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>></mo>
|
|||
|
|
<mn>0</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k>0}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27b3af208b148139eefc03f0f80fa94c38c5af45" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="k>0"></span>.<sup id="cite_ref-lyon_10-0" class="reference"><a href="#cite_note-lyon-10">[10]</a></sup> In particular, he proposed these closed-form bounds, which he proved in 2023:<sup id="cite_ref-lyon2023_11-0" class="reference"><a href="#cite_note-lyon2023-11">[11]</a></sup>
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{L\infty }(k)=2^{-1/k}(\log 2-{\frac {1}{3}}+k)\quad }">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<msub>
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>L</mi>
|
|||
|
|
<mi mathvariant="normal">∞<!-- ∞ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<msup>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>/</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>log</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mn>3</mn>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mspace width="1em" />
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \nu _{L\infty }(k)=2^{-1/k}(\log 2-{\frac {1}{3}}+k)\quad }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d2591377ce5a7b571e43130a7aa0ae5e591c8bb" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.98ex; height:5.176ex;" alt="{\displaystyle \nu _{L\infty }(k)=2^{-1/k}(\log 2-{\frac {1}{3}}+k)\quad }"></span> is a lower bound, asymptotically tight as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\to \infty }">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">→<!-- → --></mo>
|
|||
|
|
<mi mathvariant="normal">∞<!-- ∞ --></mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k\to \infty }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acf168f27ae911d0e1f71ce7e2c8fc9d1a3adeb5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.149ex; height:2.176ex;" alt="k\to \infty "></span></dd>
|
|||
|
|
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{U}(k)=2^{-1/k}(e^{-\gamma }+k)\quad }">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<msub>
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>U</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<msup>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>/</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>e</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>γ<!-- γ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mspace width="1em" />
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \nu _{U}(k)=2^{-1/k}(e^{-\gamma }+k)\quad }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed520e224ba985e574fb77b71ee5bb6883512f2" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.604ex; height:3.343ex;" alt="{\displaystyle \nu _{U}(k)=2^{-1/k}(e^{-\gamma }+k)\quad }"></span> is an upper bound, asymptotically tight as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\to 0}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">→<!-- → --></mo>
|
|||
|
|
<mn>0</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k\to 0}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fcc764df099ea8bc091518756ccc1221f6e37ba" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.988ex; height:2.176ex;" alt="{\displaystyle k\to 0}"></span></dd></dl>
|
|||
|
|
<p>Lyon also showed (informally in 2021, rigorously in 2023) two other lower bounds that are not <a href="/wiki/Closed-form_expression" title="Closed-form expression">closed-form expressions</a>, including this one involving the <a href="/wiki/Gamma_function" title="Gamma function">gamma function</a>, based on solving the integral expression substituting 1 for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-x}}">
|
|||
|
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<semantics>
|
|||
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<mrow class="MJX-TeXAtom-ORD">
|
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<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
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<msup>
|
|||
|
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<mi>e</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle e^{-x}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b201e900a30da19a1f1e4bdddcc70fe7e502be4b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.535ex; height:2.509ex;" alt="e^{-x}"></span>:
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (k)>\left({\frac {2}{\Gamma (k+1)}}\right)^{-1/k}\quad }">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>></mo>
|
|||
|
|
<msup>
|
|||
|
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<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
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<mfrac>
|
|||
|
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<mn>2</mn>
|
|||
|
|
<mrow>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
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<mrow class="MJX-TeXAtom-ORD">
|
|||
|
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<mo>/</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mrow>
|
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|
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</msup>
|
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|
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<mspace width="1em" />
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \nu (k)>\left({\frac {2}{\Gamma (k+1)}}\right)^{-1/k}\quad }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec07f2440f178b43a965d030b6d17daadbacfc2" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.418ex; height:6.843ex;" alt="{\displaystyle \nu (k)>\left({\frac {2}{\Gamma (k+1)}}\right)^{-1/k}\quad }"></span> (approaching equality as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\to 0}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">→<!-- → --></mo>
|
|||
|
|
<mn>0</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k\to 0}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fcc764df099ea8bc091518756ccc1221f6e37ba" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.988ex; height:2.176ex;" alt="{\displaystyle k\to 0}"></span>)</dd></dl>
|
|||
|
|
<p>and the tangent line at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k=1}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="k=1"></span> where the derivative was found to be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu ^{\prime }(1)\approx 0.9680448}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<msup>
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>≈<!-- ≈ --></mo>
|
|||
|
|
<mn>0.9680448</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \nu ^{\prime }(1)\approx 0.9680448}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad24afe34a23eb1560b491689eec1e7dea040c63" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.959ex; height:3.009ex;" alt="{\displaystyle \nu ^{\prime }(1)\approx 0.9680448}"></span>:
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (k)\geq \nu (1)+(k-1)\nu ^{\prime }(1)\quad }">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>≥<!-- ≥ --></mo>
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mspace width="1em" />
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \nu (k)\geq \nu (1)+(k-1)\nu ^{\prime }(1)\quad }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c4d3c22d938a07214c417f6975f7cbd17fbf1a9" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.655ex; height:3.009ex;" alt="{\displaystyle \nu (k)\geq \nu (1)+(k-1)\nu ^{\prime }(1)\quad }"></span> (with equality at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k=1}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="k=1"></span>)</dd>
|
|||
|
|
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (k)\geq \log(2)+(k-1)(\gamma -2\operatorname {Ei} (-\log 2)-\log \log 2)}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>≥<!-- ≥ --></mo>
|
|||
|
|
<mi>log</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>γ<!-- γ --></mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
<mi>Ei</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>log</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>log</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mi>log</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \nu (k)\geq \log(2)+(k-1)(\gamma -2\operatorname {Ei} (-\log 2)-\log \log 2)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/478102c9fbf3107ca2ffb3e6bae6835a84d93f3a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.096ex; height:2.843ex;" alt="{\displaystyle \nu (k)\geq \log(2)+(k-1)(\gamma -2\operatorname {Ei} (-\log 2)-\log \log 2)}"></span></dd></dl>
|
|||
|
|
<p>where Ei is the <a href="/wiki/Exponential_integral" title="Exponential integral">exponential integral</a>.<sup id="cite_ref-lyon_10-1" class="reference"><a href="#cite_note-lyon-10">[10]</a></sup><sup id="cite_ref-lyon2023_11-1" class="reference"><a href="#cite_note-lyon2023-11">[11]</a></sup>
|
|||
|
|
</p><p>Additionally, he showed that interpolations between bounds could provide excellent approximations or tighter bounds to the median, including an approximation that is exact at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k=1}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="k=1"></span> (where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (1)=\log 2}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>log</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \nu (1)=\log 2}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f48b9898b0be0be38bbdfc4049164982bce09ea3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.824ex; height:2.843ex;" alt="{\displaystyle \nu (1)=\log 2}"></span>) and has a maximum relative error less than 0.6%. Interpolated approximations and bounds are all of the form
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (k)\approx {\tilde {g}}(k)\nu _{L\infty }(k)+(1-{\tilde {g}}(k))\nu _{U}(k)}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>≈<!-- ≈ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>g</mi>
|
|||
|
|
<mo stretchy="false">~<!-- ~ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>L</mi>
|
|||
|
|
<mi mathvariant="normal">∞<!-- ∞ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>g</mi>
|
|||
|
|
<mo stretchy="false">~<!-- ~ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>U</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \nu (k)\approx {\tilde {g}}(k)\nu _{L\infty }(k)+(1-{\tilde {g}}(k))\nu _{U}(k)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8435e11856bb84972296cd69082bd5ac49609938" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.333ex; height:2.843ex;" alt="{\displaystyle \nu (k)\approx {\tilde {g}}(k)\nu _{L\infty }(k)+(1-{\tilde {g}}(k))\nu _{U}(k)}"></span></dd></dl>
|
|||
|
|
<p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {g}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>g</mi>
|
|||
|
|
<mo stretchy="false">~<!-- ~ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\tilde {g}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcf709e979316ee494a3f076f7e1d97be44a3f8f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.232ex; height:2.509ex;" alt="{\tilde {g}}"></span> is an interpolating function running monotonically from 0 at low <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="k"></span> to 1 at high <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="k"></span>, approximating an ideal, or exact, interpolator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(k)}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>g</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle g(k)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77b991fb29325fbaa24480e075e54871f5128a62" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="g(k)"></span>:
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(k)={\frac {\nu _{U}(k)-\nu (k)}{\nu _{U}(k)-\nu _{L\infty }(k)}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>g</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mrow>
|
|||
|
|
<msub>
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>U</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<msub>
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>U</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>L</mi>
|
|||
|
|
<mi mathvariant="normal">∞<!-- ∞ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle g(k)={\frac {\nu _{U}(k)-\nu (k)}{\nu _{U}(k)-\nu _{L\infty }(k)}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ea545145faf7173d894259d15c1b2ed59cf70d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.737ex; height:6.509ex;" alt="{\displaystyle g(k)={\frac {\nu _{U}(k)-\nu (k)}{\nu _{U}(k)-\nu _{L\infty }(k)}}}"></span></dd></dl>
|
|||
|
|
<p>For the simplest interpolating function considered, a first-order rational function
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {g}}_{1}(k)={\frac {k}{b_{0}+k}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<msub>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>g</mi>
|
|||
|
|
<mo stretchy="false">~<!-- ~ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mrow>
|
|||
|
|
<msub>
|
|||
|
|
<mi>b</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>0</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\tilde {g}}_{1}(k)={\frac {k}{b_{0}+k}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6e65ab5d33035c86360b97f1dde6799ddeec9d4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.344ex; height:5.843ex;" alt="{\displaystyle {\tilde {g}}_{1}(k)={\frac {k}{b_{0}+k}}}"></span></dd></dl>
|
|||
|
|
<p>the tightest lower bound has
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}={\frac {{\frac {8}{405}}+e^{-\gamma }\log 2-{\frac {\log ^{2}2}{2}}}{e^{-\gamma }-\log 2+{\frac {1}{3}}}}-\log 2\approx 0.143472}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<msub>
|
|||
|
|
<mi>b</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>0</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>8</mn>
|
|||
|
|
<mn>405</mn>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>e</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>γ<!-- γ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mi>log</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mrow>
|
|||
|
|
<msup>
|
|||
|
|
<mi>log</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
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|
|||
|
|
</msup>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<msup>
|
|||
|
|
<mi>e</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>γ<!-- γ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>log</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mn>3</mn>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>log</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
<mo>≈<!-- ≈ --></mo>
|
|||
|
|
<mn>0.143472</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle b_{0}={\frac {{\frac {8}{405}}+e^{-\gamma }\log 2-{\frac {\log ^{2}2}{2}}}{e^{-\gamma }-\log 2+{\frac {1}{3}}}}-\log 2\approx 0.143472}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e4d17f6dff2d8b4188f7d396887129affd9edb8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:47.587ex; height:8.843ex;" alt="{\displaystyle b_{0}={\frac {{\frac {8}{405}}+e^{-\gamma }\log 2-{\frac {\log ^{2}2}{2}}}{e^{-\gamma }-\log 2+{\frac {1}{3}}}}-\log 2\approx 0.143472}"></span></dd></dl>
|
|||
|
|
<p>and the tightest upper bound has
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}={\frac {e^{-\gamma }-\log 2+{\frac {1}{3}}}{1-{\frac {e^{-\gamma }\pi ^{2}}{12}}}}\approx 0.374654}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<msub>
|
|||
|
|
<mi>b</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>0</mn>
|
|||
|
|
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|
|||
|
|
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|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
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|
|||
|
|
<msup>
|
|||
|
|
<mi>e</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>γ<!-- γ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>log</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mn>3</mn>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mrow>
|
|||
|
|
<msup>
|
|||
|
|
<mi>e</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>γ<!-- γ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<msup>
|
|||
|
|
<mi>π<!-- π --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mrow>
|
|||
|
|
<mn>12</mn>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>≈<!-- ≈ --></mo>
|
|||
|
|
<mn>0.374654</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle b_{0}={\frac {e^{-\gamma }-\log 2+{\frac {1}{3}}}{1-{\frac {e^{-\gamma }\pi ^{2}}{12}}}}\approx 0.374654}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8762f4ac41b74d0ac8c557125bda65e5431ee804" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:33.216ex; height:8.343ex;" alt="{\displaystyle b_{0}={\frac {e^{-\gamma }-\log 2+{\frac {1}{3}}}{1-{\frac {e^{-\gamma }\pi ^{2}}{12}}}}\approx 0.374654}"></span></dd></dl>
|
|||
|
|
<p>The interpolated bounds are plotted (mostly inside the yellow region) in the <a href="/wiki/Log%E2%80%93log_plot" title="Log–log plot">log–log plot</a> shown. Even tighter bounds are available using different interpolating functions, but not usually with closed-form parameters like these.<sup id="cite_ref-lyon_10-2" class="reference"><a href="#cite_note-lyon-10">[10]</a></sup>
|
|||
|
|
</p>
|
|||
|
|
<h3><span class="mw-headline" id="Summation">Summation</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=9" title="Edit section: Summation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
|
|||
|
|
<p>If <i>X</i><sub><i>i</i></sub> has a Gamma(<i>k</i><sub><i>i</i></sub>, <i>θ</i>) distribution for <i>i</i> = 1, 2, ..., <i>N</i> (i.e., all distributions have the same scale parameter <i>θ</i>), then
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{N}X_{i}\sim \mathrm {Gamma} \left(\sum _{i=1}^{N}k_{i},\theta \right)}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<munderover>
|
|||
|
|
<mo>∑<!-- ∑ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</munderover>
|
|||
|
|
<msub>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>∼<!-- ∼ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi mathvariant="normal">G</mi>
|
|||
|
|
<mi mathvariant="normal">a</mi>
|
|||
|
|
<mi mathvariant="normal">m</mi>
|
|||
|
|
<mi mathvariant="normal">m</mi>
|
|||
|
|
<mi mathvariant="normal">a</mi>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<mrow>
|
|||
|
|
<munderover>
|
|||
|
|
<mo>∑<!-- ∑ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</munderover>
|
|||
|
|
<msub>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{N}X_{i}\sim \mathrm {Gamma} \left(\sum _{i=1}^{N}k_{i},\theta \right)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62d35fd0b7cd574ea4f7a49eb96f107f6ffe5edb" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.531ex; height:7.509ex;" alt=" \sum_{i=1}^N X_i \sim\mathrm{Gamma} \left( \sum_{i=1}^N k_i, \theta \right)"></span></dd></dl>
|
|||
|
|
<p>provided all <i>X</i><sub><i>i</i></sub> are <a href="/wiki/Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a>.
|
|||
|
|
</p><p>For the cases where the <i>X</i><sub><i>i</i></sub> are <a href="/wiki/Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a> but have different scale parameters, see Mathai <sup id="cite_ref-12" class="reference"><a href="#cite_note-12">[12]</a></sup> or Moschopoulos.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13">[13]</a></sup>
|
|||
|
|
</p><p>The gamma distribution exhibits <a href="/wiki/Infinite_divisibility_(probability)" title="Infinite divisibility (probability)">infinite divisibility</a>.
|
|||
|
|
</p>
|
|||
|
|
<h3><span class="mw-headline" id="Scaling">Scaling</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=10" title="Edit section: Scaling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
|
|||
|
|
<p>If
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \mathrm {Gamma} (k,\theta ),}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mo>∼<!-- ∼ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi mathvariant="normal">G</mi>
|
|||
|
|
<mi mathvariant="normal">a</mi>
|
|||
|
|
<mi mathvariant="normal">m</mi>
|
|||
|
|
<mi mathvariant="normal">m</mi>
|
|||
|
|
<mi mathvariant="normal">a</mi>
|
|||
|
|
</mrow>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle X\sim \mathrm {Gamma} (k,\theta ),}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/391514411e8eaf5fdc115ca83083262b1fc48424" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.891ex; height:2.843ex;" alt="X \sim \mathrm{Gamma}(k, \theta),"></span></dd></dl>
|
|||
|
|
<p>then, for any <i>c</i> > 0,
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle cX\sim \mathrm {Gamma} (k,c\,\theta ),}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>c</mi>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mo>∼<!-- ∼ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi mathvariant="normal">G</mi>
|
|||
|
|
<mi mathvariant="normal">a</mi>
|
|||
|
|
<mi mathvariant="normal">m</mi>
|
|||
|
|
<mi mathvariant="normal">m</mi>
|
|||
|
|
<mi mathvariant="normal">a</mi>
|
|||
|
|
</mrow>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>c</mi>
|
|||
|
|
<mspace width="thinmathspace" />
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle cX\sim \mathrm {Gamma} (k,c\,\theta ),}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/615789eea5ea246ac96af8627865abbad8f1b404" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.292ex; height:2.843ex;" alt="{\displaystyle cX\sim \mathrm {Gamma} (k,c\,\theta ),}"></span> by moment generating functions,</dd></dl>
|
|||
|
|
<p>or equivalently, if
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \mathrm {Gamma} \left(\alpha ,\beta \right)}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mo>∼<!-- ∼ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi mathvariant="normal">G</mi>
|
|||
|
|
<mi mathvariant="normal">a</mi>
|
|||
|
|
<mi mathvariant="normal">m</mi>
|
|||
|
|
<mi mathvariant="normal">m</mi>
|
|||
|
|
<mi mathvariant="normal">a</mi>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle X\sim \mathrm {Gamma} \left(\alpha ,\beta \right)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3247407b9d18e6a48db04762641c9887835677c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.149ex; height:2.843ex;" alt="{\displaystyle X\sim \mathrm {Gamma} \left(\alpha ,\beta \right)}"></span> (shape-rate parameterization)</dd></dl>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle cX\sim \mathrm {Gamma} \left(\alpha ,{\frac {\beta }{c}}\right),}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>c</mi>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mo>∼<!-- ∼ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi mathvariant="normal">G</mi>
|
|||
|
|
<mi mathvariant="normal">a</mi>
|
|||
|
|
<mi mathvariant="normal">m</mi>
|
|||
|
|
<mi mathvariant="normal">m</mi>
|
|||
|
|
<mi mathvariant="normal">a</mi>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mi>c</mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle cX\sim \mathrm {Gamma} \left(\alpha ,{\frac {\beta }{c}}\right),}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9d2967d0b2f02976cb6572d8be0bb3281711d7e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.638ex; height:6.176ex;" alt="{\displaystyle cX\sim \mathrm {Gamma} \left(\alpha ,{\frac {\beta }{c}}\right),}"></span></dd></dl>
|
|||
|
|
<p>Indeed, we know that if <i>X</i> is an <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential r.v.</a> with rate <i>λ</i>, then <i>cX</i> is an exponential r.v. with rate <i>λ</i>/<i>c</i>; the same thing is valid with Gamma variates (and this can be checked using the <a href="/wiki/Moment-generating_function" title="Moment-generating function">moment-generating function</a>, see, e.g.,<a rel="nofollow" class="external text" href="http://www.stat.washington.edu/thompson/S341_10/Notes/week4.pdf">these notes</a>, 10.4-(ii)): multiplication by a positive constant <i>c</i> divides the rate (or, equivalently, multiplies the scale).
|
|||
|
|
</p>
|
|||
|
|
<h3><span class="mw-headline" id="Exponential_family">Exponential family</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=11" title="Edit section: Exponential family"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
|
|||
|
|
<p>The gamma distribution is a two-parameter <a href="/wiki/Exponential_family" title="Exponential family">exponential family</a> with <a href="/wiki/Natural_parameters" class="mw-redirect" title="Natural parameters">natural parameters</a> <i>k</i> − 1 and −1/<i>θ</i> (equivalently, <i>α</i> − 1 and −<i>β</i>), and <a href="/wiki/Natural_statistics" class="mw-redirect" title="Natural statistics">natural statistics</a> <i>X</i> and ln(<i>X</i>).
|
|||
|
|
</p><p>If the shape parameter <i>k</i> is held fixed, the resulting one-parameter family of distributions is a <a href="/wiki/Natural_exponential_family" title="Natural exponential family">natural exponential family</a>.
|
|||
|
|
</p>
|
|||
|
|
<h3><span class="mw-headline" id="Logarithmic_expectation_and_variance">Logarithmic expectation and variance</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=12" title="Edit section: Logarithmic expectation and variance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
|
|||
|
|
<p>One can show that
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [\ln(X)]=\psi (\alpha )-\ln(\beta )}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi mathvariant="normal">E</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">[</mo>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo stretchy="false">]</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>ψ<!-- ψ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [\ln(X)]=\psi (\alpha )-\ln(\beta )}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6da14ff7ed563c7e86154998ef6fd180e79c9bfa" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.435ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [\ln(X)]=\psi (\alpha )-\ln(\beta )}"></span></dd></dl>
|
|||
|
|
<p>or equivalently,
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [\ln(X)]=\psi (k)+\ln(\theta )}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi mathvariant="normal">E</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">[</mo>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo stretchy="false">]</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>ψ<!-- ψ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [\ln(X)]=\psi (k)+\ln(\theta )}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/186737f3b184bf00519b3a4b1412a560e1216093" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.917ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [\ln(X)]=\psi (k)+\ln(\theta )}"></span></dd></dl>
|
|||
|
|
<p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>ψ<!-- ψ --></mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="\psi "></span> is the <a href="/wiki/Digamma_function" title="Digamma function">digamma function</a>. Likewise,
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {var} [\ln(X)]=\psi ^{(1)}(\alpha )=\psi ^{(1)}(k)}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>var</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">[</mo>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo stretchy="false">]</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>ψ<!-- ψ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>ψ<!-- ψ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \operatorname {var} [\ln(X)]=\psi ^{(1)}(\alpha )=\psi ^{(1)}(k)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b193ce127d5d0de9a3430b7dc803c092262f7b5c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.532ex; height:3.343ex;" alt="{\displaystyle \operatorname {var} [\ln(X)]=\psi ^{(1)}(\alpha )=\psi ^{(1)}(k)}"></span></dd></dl>
|
|||
|
|
<p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{(1)}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<msup>
|
|||
|
|
<mi>ψ<!-- ψ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \psi ^{(1)}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f824d0b8ec54f716ece0f1d8acf4898cab2f7dd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.847ex; height:3.176ex;" alt="{\displaystyle \psi ^{(1)}}"></span> is the <a href="/wiki/Trigamma_function" title="Trigamma function">trigamma function</a>.
|
|||
|
|
</p><p>This can be derived using the <a href="/wiki/Exponential_family" title="Exponential family">exponential family</a> formula for the <a href="/wiki/Exponential_family#Moment_generating_function_of_the_sufficient_statistic" title="Exponential family">moment generating function of the sufficient statistic</a>, because one of the sufficient statistics of the gamma distribution is ln(<i>x</i>).
|
|||
|
|
</p>
|
|||
|
|
<h3><span class="mw-headline" id="Information_entropy">Information entropy</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=13" title="Edit section: Information entropy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
|
|||
|
|
<p>The <a href="/wiki/Information_entropy" class="mw-redirect" title="Information entropy">information entropy</a> is
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {H} (X)&=\operatorname {E} [-\ln(p(X))]\\[4pt]&=\operatorname {E} [-\alpha \ln(\beta )+\ln(\Gamma (\alpha ))-(\alpha -1)\ln(X)+\beta X]\\[4pt]&=\alpha -\ln(\beta )+\ln(\Gamma (\alpha ))+(1-\alpha )\psi (\alpha ).\end{aligned}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
|
|||
|
|
<mtr>
|
|||
|
|
<mtd>
|
|||
|
|
<mi mathvariant="normal">H</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mtd>
|
|||
|
|
<mtd>
|
|||
|
|
<mi></mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi mathvariant="normal">E</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">[</mo>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>p</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo stretchy="false">]</mo>
|
|||
|
|
</mtd>
|
|||
|
|
</mtr>
|
|||
|
|
<mtr>
|
|||
|
|
<mtd />
|
|||
|
|
<mtd>
|
|||
|
|
<mi></mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi mathvariant="normal">E</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">[</mo>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>+</mo>
|
|||
|
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|
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|
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|
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|
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|
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|
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<mi>ln</mi>
|
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|
|||
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|
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|
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|
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|
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<mi>ln</mi>
|
|||
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<mo>⁡<!-- --></mo>
|
|||
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|
|||
|
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<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
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<mo stretchy="false">(</mo>
|
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<mi>α<!-- α --></mi>
|
|||
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<mo stretchy="false">)</mo>
|
|||
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|
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|
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|
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<mn>1</mn>
|
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|
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|
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|
|||
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<mi>ψ<!-- ψ --></mi>
|
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|
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|
|||
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|
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|
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|
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|
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|
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|
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|
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {H} (X)&=\operatorname {E} [-\ln(p(X))]\\[4pt]&=\operatorname {E} [-\alpha \ln(\beta )+\ln(\Gamma (\alpha ))-(\alpha -1)\ln(X)+\beta X]\\[4pt]&=\alpha -\ln(\beta )+\ln(\Gamma (\alpha ))+(1-\alpha )\psi (\alpha ).\end{aligned}}}</annotation>
|
|||
|
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</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37a24251136eb110aea24081dcffb2ee9e9648d8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:54.769ex; height:11.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {H} (X)&=\operatorname {E} [-\ln(p(X))]\\[4pt]&=\operatorname {E} [-\alpha \ln(\beta )+\ln(\Gamma (\alpha ))-(\alpha -1)\ln(X)+\beta X]\\[4pt]&=\alpha -\ln(\beta )+\ln(\Gamma (\alpha ))+(1-\alpha )\psi (\alpha ).\end{aligned}}}"></span></dd></dl>
|
|||
|
|
<p>In the <i>k</i>, <i>θ</i> parameterization, the <a href="/wiki/Information_entropy" class="mw-redirect" title="Information entropy">information entropy</a> is given by
|
|||
|
|
</p>
|
|||
|
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<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {H} (X)=k+\ln(\theta )+\ln(\Gamma (k))+(1-k)\psi (k).}">
|
|||
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|||
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|
|||
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|||
|
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|
|||
|
|
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|
|||
|
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|
|||
|
|
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|
|||
|
|
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|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \operatorname {H} (X)=k+\ln(\theta )+\ln(\Gamma (k))+(1-k)\psi (k).}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/232ab16ba751c3d274b432ed4698d7babc031288" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.628ex; height:2.843ex;" alt="\operatorname{H}(X) =k + \ln(\theta) + \ln(\Gamma(k)) + (1-k)\psi(k)."></span></dd></dl>
|
|||
|
|
<h3><span id="Kullback.E2.80.93Leibler_divergence"></span><span class="mw-headline" id="Kullback–Leibler_divergence">Kullback–Leibler divergence</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=14" title="Edit section: Kullback–Leibler divergence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
|
|||
|
|
<figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Gamma-KL-3D.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Gamma-KL-3D.png/320px-Gamma-KL-3D.png" decoding="async" width="320" height="209" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Gamma-KL-3D.png/480px-Gamma-KL-3D.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Gamma-KL-3D.png/640px-Gamma-KL-3D.png 2x" data-file-width="2048" data-file-height="1339" /></a><figcaption>Illustration of the Kullback–Leibler (KL) divergence for two gamma PDFs. Here <i>β</i> = <i>β</i><sub>0</sub> + 1 which are set to 1, 2, 3, 4, 5 and 6. The typical asymmetry for the KL divergence is clearly visible.</figcaption></figure>
|
|||
|
|
<p>The <a href="/wiki/Kullback%E2%80%93Leibler_divergence" title="Kullback–Leibler divergence">Kullback–Leibler divergence</a> (KL-divergence), of Gamma(<i>α</i><sub><i>p</i></sub>, <i>β</i><sub><i>p</i></sub>) ("true" distribution) from Gamma(<i>α</i><sub><i>q</i></sub>, <i>β</i><sub><i>q</i></sub>) ("approximating" distribution) is given by<sup id="cite_ref-14" class="reference"><a href="#cite_note-14">[14]</a></sup>
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}D_{\mathrm {KL} }(\alpha _{p},\beta _{p};\alpha _{q},\beta _{q})={}&(\alpha _{p}-\alpha _{q})\psi (\alpha _{p})-\log \Gamma (\alpha _{p})+\log \Gamma (\alpha _{q})\\&{}+\alpha _{q}(\log \beta _{p}-\log \beta _{q})+\alpha _{p}{\frac {\beta _{q}-\beta _{p}}{\beta _{p}}}.\end{aligned}}}">
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|||
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|
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|
|||
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|
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|
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|
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|
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|
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|
|||
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|
|||
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|
|||
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|
|||
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<mo stretchy="false">(</mo>
|
|||
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|
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|
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|
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|
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|
|||
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}D_{\mathrm {KL} }(\alpha _{p},\beta _{p};\alpha _{q},\beta _{q})={}&(\alpha _{p}-\alpha _{q})\psi (\alpha _{p})-\log \Gamma (\alpha _{p})+\log \Gamma (\alpha _{q})\\&{}+\alpha _{q}(\log \beta _{p}-\log \beta _{q})+\alpha _{p}{\frac {\beta _{q}-\beta _{p}}{\beta _{p}}}.\end{aligned}}}</annotation>
|
|||
|
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</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5567126eea1d9b2264fbc9b966d3621a739c5de" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:62.413ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}D_{\mathrm {KL} }(\alpha _{p},\beta _{p};\alpha _{q},\beta _{q})={}&(\alpha _{p}-\alpha _{q})\psi (\alpha _{p})-\log \Gamma (\alpha _{p})+\log \Gamma (\alpha _{q})\\&{}+\alpha _{q}(\log \beta _{p}-\log \beta _{q})+\alpha _{p}{\frac {\beta _{q}-\beta _{p}}{\beta _{p}}}.\end{aligned}}}"></span></dd></dl>
|
|||
|
|
<p>Written using the <i>k</i>, <i>θ</i> parameterization, the KL-divergence of Gamma(<i>k<sub>p</sub></i>, θ<sub><i>p</i></sub>) from Gamma(<i>k<sub>q</sub></i>, <i>θ</i><sub><i>q</i></sub>) is given by
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}D_{\mathrm {KL} }(k_{p},\theta _{p};k_{q},\theta _{q})={}&(k_{p}-k_{q})\psi (k_{p})-\log \Gamma (k_{p})+\log \Gamma (k_{q})\\&{}+k_{q}(\log \theta _{q}-\log \theta _{p})+k_{p}{\frac {\theta _{p}-\theta _{q}}{\theta _{q}}}.\end{aligned}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
|
|||
|
|
<mtr>
|
|||
|
|
<mtd>
|
|||
|
|
<msub>
|
|||
|
|
<mi>D</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi mathvariant="normal">K</mi>
|
|||
|
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<mi mathvariant="normal">L</mi>
|
|||
|
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</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
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<mi>p</mi>
|
|||
|
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</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>,</mo>
|
|||
|
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<msub>
|
|||
|
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<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
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<mi>p</mi>
|
|||
|
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</mrow>
|
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</msub>
|
|||
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<mo>;</mo>
|
|||
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<msub>
|
|||
|
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<mi>k</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
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<mi>q</mi>
|
|||
|
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</mrow>
|
|||
|
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</msub>
|
|||
|
|
<mo>,</mo>
|
|||
|
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<msub>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
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<mi>q</mi>
|
|||
|
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</mrow>
|
|||
|
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</msub>
|
|||
|
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<mo stretchy="false">)</mo>
|
|||
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<mo>=</mo>
|
|||
|
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<mrow class="MJX-TeXAtom-ORD">
|
|||
|
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|
|||
|
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</mrow>
|
|||
|
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</mtd>
|
|||
|
|
<mtd>
|
|||
|
|
<mi></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
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<mi>p</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>q</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mi>ψ<!-- ψ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>p</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>log</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>p</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mi>log</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>q</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mtd>
|
|||
|
|
</mtr>
|
|||
|
|
<mtr>
|
|||
|
|
<mtd />
|
|||
|
|
<mtd>
|
|||
|
|
<mi></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
|
|||
|
|
</mrow>
|
|||
|
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<mo>+</mo>
|
|||
|
|
<msub>
|
|||
|
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<mi>k</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
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<mi>q</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>log</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>q</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>log</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>p</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>p</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
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<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mrow>
|
|||
|
|
<msub>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>p</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>q</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
</mrow>
|
|||
|
|
<msub>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>q</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>.</mo>
|
|||
|
|
</mtd>
|
|||
|
|
</mtr>
|
|||
|
|
</mtable>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}D_{\mathrm {KL} }(k_{p},\theta _{p};k_{q},\theta _{q})={}&(k_{p}-k_{q})\psi (k_{p})-\log \Gamma (k_{p})+\log \Gamma (k_{q})\\&{}+k_{q}(\log \theta _{q}-\log \theta _{p})+k_{p}{\frac {\theta _{p}-\theta _{q}}{\theta _{q}}}.\end{aligned}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bae9104706395e7b3d9da9f41c6b7a69b8d9423" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:60.028ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}D_{\mathrm {KL} }(k_{p},\theta _{p};k_{q},\theta _{q})={}&(k_{p}-k_{q})\psi (k_{p})-\log \Gamma (k_{p})+\log \Gamma (k_{q})\\&{}+k_{q}(\log \theta _{q}-\log \theta _{p})+k_{p}{\frac {\theta _{p}-\theta _{q}}{\theta _{q}}}.\end{aligned}}}"></span></dd></dl>
|
|||
|
|
<h3><span class="mw-headline" id="Laplace_transform">Laplace transform</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=15" title="Edit section: Laplace transform"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
|
|||
|
|
<p>The <a href="/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a> of the gamma PDF is
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(s)=(1+\theta s)^{-k}={\frac {\beta ^{\alpha }}{(s+\beta )^{\alpha }}}.}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>F</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>s</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mi>s</mi>
|
|||
|
|
<msup>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<msup>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mrow>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>s</mi>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<msup>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>.</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle F(s)=(1+\theta s)^{-k}={\frac {\beta ^{\alpha }}{(s+\beta )^{\alpha }}}.}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81d5c9dac2cf6d0ae13d59b77058972caa492522" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.037ex; height:6.176ex;" alt="F(s) = (1 + \theta s)^{-k} = \frac{\beta^\alpha}{(s + \beta)^\alpha} ."></span></dd></dl>
|
|||
|
|
<h2><span class="mw-headline" id="Related_distributions">Related distributions</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=16" title="Edit section: Related distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
|
|||
|
|
<h3><span class="mw-headline" id="General">General</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=17" title="Edit section: General"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
|
|||
|
|
<ul><li>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},\ldots ,X_{n}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<msub>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mo>…<!-- … --></mo>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>n</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},\ldots ,X_{n}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d67872301909a9d739e265252ad0c7339cead069" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.312ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},\ldots ,X_{n}}"></span> be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>n</mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="n"></span> independent and identically distributed random variables following an <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a> with rate parameter λ, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}X_{i}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<munder>
|
|||
|
|
<mo>∑<!-- ∑ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</munder>
|
|||
|
|
<msub>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \sum _{i}X_{i}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1973eff94821789d2a7ff09c2f5dc31be87f88d7" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:6.466ex; height:5.509ex;" alt="{\displaystyle \sum _{i}X_{i}}"></span> ~ Gamma(n, λ) where n is the shape parameter and <i>λ</i> is the rate, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\bar {X}}={\frac {1}{n}}\sum _{i}X_{i}\sim \operatorname {Gamma} (n,n\lambda )}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="false" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mo stretchy="false">¯<!-- ¯ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mi>n</mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<munder>
|
|||
|
|
<mo>∑<!-- ∑ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</munder>
|
|||
|
|
<msub>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>∼<!-- ∼ --></mo>
|
|||
|
|
<mi>Gamma</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>n</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>n</mi>
|
|||
|
|
<mi>λ<!-- λ --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\textstyle {\bar {X}}={\frac {1}{n}}\sum _{i}X_{i}\sim \operatorname {Gamma} (n,n\lambda )}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57802ca2bafc92f75dfc34843c58864d2d145931" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.76ex; height:3.343ex;" alt="{\textstyle {\bar {X}}={\frac {1}{n}}\sum _{i}X_{i}\sim \operatorname {Gamma} (n,n\lambda )}"></span>.</li>
|
|||
|
|
<li>If <i>X</i> ~ Gamma(1, <i>λ</i>) (in the shape–rate parametrization), then <i>X</i> has an <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a> with rate parameter <i>λ</i>. In the shape-scale parametrization, <i>X</i> ~ Gamma(1, <i>λ</i>) has an exponential distribution with rate parameter 1/<i>λ</i>.</li>
|
|||
|
|
<li>If <i>X</i> ~ Gamma(<i>ν</i>/2, 2) (in the shape–scale parametrization), then <i>X</i> is identical to <i>χ</i><sup>2</sup>(<i>ν</i>), the <a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">chi-squared distribution</a> with <i>ν</i> degrees of freedom. Conversely, if <i>Q</i> ~ <i>χ</i><sup>2</sup>(<i>ν</i>) and <i>c</i> is a positive constant, then <i>cQ</i> ~ Gamma(<i>ν</i>/2, 2<i>c</i>).</li>
|
|||
|
|
<li>If <i>θ=1/k</i>, one obtains the <a href="/wiki/Schulz-Zimm_distribution" class="mw-redirect" title="Schulz-Zimm distribution">Schulz-Zimm distribution</a>, which is most prominently used to model polymer chain lengths.</li>
|
|||
|
|
<li>If <i>k</i> is an <a href="/wiki/Integer" title="Integer">integer</a>, the gamma distribution is an <a href="/wiki/Erlang_distribution" title="Erlang distribution">Erlang distribution</a> and is the probability distribution of the waiting time until the <i>k</i>th "arrival" in a one-dimensional <a href="/wiki/Poisson_process" class="mw-redirect" title="Poisson process">Poisson process</a> with intensity 1/<i>θ</i>. If</li></ul>
|
|||
|
|
<dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \Gamma (k\in \mathbf {Z} ,\theta ),\qquad Y\sim \operatorname {Pois} \left({\frac {x}{\theta }}\right),}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mo>∼<!-- ∼ --></mo>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>∈<!-- ∈ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi mathvariant="bold">Z</mi>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mspace width="2em" />
|
|||
|
|
<mi>Y</mi>
|
|||
|
|
<mo>∼<!-- ∼ --></mo>
|
|||
|
|
<mi>Pois</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle X\sim \Gamma (k\in \mathbf {Z} ,\theta ),\qquad Y\sim \operatorname {Pois} \left({\frac {x}{\theta }}\right),}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cea8fc216dc5113f4d745dcee1fd769c4608dff6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:36.598ex; height:5.009ex;" alt="{\displaystyle X\sim \Gamma (k\in \mathbf {Z} ,\theta ),\qquad Y\sim \operatorname {Pois} \left({\frac {x}{\theta }}\right),}"></span></dd></dl></dd>
|
|||
|
|
<dd>then
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X>x)=P(Y<k).}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>P</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mo>></mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>P</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>Y</mi>
|
|||
|
|
<mo><</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>.</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle P(X>x)=P(Y<k).}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcd3aee7ab286228c1c22b1ad3371d3f9aa2538a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.346ex; height:2.843ex;" alt="P(X > x) = P(Y < k)."></span></dd></dl></dd></dl>
|
|||
|
|
<ul><li>If <i>X</i> has a <a href="/wiki/Maxwell%E2%80%93Boltzmann_distribution" title="Maxwell–Boltzmann distribution">Maxwell–Boltzmann distribution</a> with parameter <i>a</i>, then</li></ul>
|
|||
|
|
<dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{2}\sim \Gamma \left({\frac {3}{2}},2a^{2}\right).}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<msup>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo>∼<!-- ∼ --></mo>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>3</mn>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
<msup>
|
|||
|
|
<mi>a</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>.</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle X^{2}\sim \Gamma \left({\frac {3}{2}},2a^{2}\right).}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/519b56baa947c79685e8946bb422b39654551779" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.924ex; height:6.176ex;" alt="{\displaystyle X^{2}\sim \Gamma \left({\frac {3}{2}},2a^{2}\right).}"></span></dd></dl></dd></dl>
|
|||
|
|
<ul><li>If <i>X</i> ~ Gamma(<i>k</i>, <i>θ</i>), then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \log X}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="false" scriptlevel="0">
|
|||
|
|
<mi>log</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\textstyle \log X}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b112df3c2dc3c9c78d5b897a9d195737fd12371a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.339ex; height:2.509ex;" alt="{\textstyle \log X}"></span> follows an exponential-gamma (abbreviated exp-gamma) distribution.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15">[15]</a></sup> It is sometimes referred to as the log-gamma distribution.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16">[16]</a></sup> Formulas for its mean and variance are in the section <a href="#Logarithmic_expectation_and_variance">#Logarithmic expectation and variance</a>.</li>
|
|||
|
|
<li>If <i>X</i> ~ Gamma(<i>k</i>, <i>θ</i>), then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {X}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<msqrt>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
</msqrt>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\sqrt {X}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88a98f7181b035dbddcd947c5f1408b9c5ed23cd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.916ex; height:3.009ex;" alt="\sqrt{X}"></span> follows a <a href="/wiki/Generalized_gamma_distribution" title="Generalized gamma distribution">generalized gamma distribution</a> with parameters <i>p</i> = 2, <i>d</i> = 2<i>k</i>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\sqrt {\theta }}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>a</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<msqrt>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
</msqrt>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle a={\sqrt {\theta }}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1d6f6db29b41fef2dcdc6f1cfee82644296c9e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.355ex; height:3.009ex;" alt="a = \sqrt{\theta}"></span><sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2012)">citation needed</span></a></i>]</sup>.</li>
|
|||
|
|
<li>More generally, if <i>X</i> ~ Gamma(<i>k</i>,<i>θ</i>), then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{q}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<msup>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>q</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle X^{q}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ff55e24541193805d60671f025c948707ce9933" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.985ex; height:2.343ex;" alt="X^{q}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q>0}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>q</mi>
|
|||
|
|
<mo>></mo>
|
|||
|
|
<mn>0</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle q>0}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/482e0a33d9e8fd6307b5f68a5182c2d0d14efc9c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="q>0"></span> follows a <a href="/wiki/Generalized_gamma_distribution" title="Generalized gamma distribution">generalized gamma distribution</a> with parameters <i>p</i> = 1/<i>q</i>, <i>d</i> = <i>k</i>/<i>q</i>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=\theta ^{q}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>a</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>q</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle a=\theta ^{q}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/142cecd27f4bc31247ef7624e1ab3c509f343039" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.407ex; height:2.343ex;" alt="a=\theta ^{q}"></span>.</li>
|
|||
|
|
<li>If <i>X</i> ~ Gamma(<i>k</i>, <i>θ</i>) with shape <i>k</i> and scale <i>θ</i>, then 1/<i>X</i> ~ Inv-Gamma(<i>k</i>, <i>θ</i><sup>−1</sup>) (see <a href="/wiki/Inverse-gamma_distribution" title="Inverse-gamma distribution">Inverse-gamma distribution</a> for derivation).</li>
|
|||
|
|
<li>Parametrization 1: If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}\sim \Gamma (\alpha _{k},\theta _{k})\,}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<msub>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>∼<!-- ∼ --></mo>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mspace width="thinmathspace" />
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle X_{k}\sim \Gamma (\alpha _{k},\theta _{k})\,}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1f4bce9ca7bb5b08324fd5622ccee7d524aed9e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.55ex; height:2.843ex;" alt="{\displaystyle X_{k}\sim \Gamma (\alpha _{k},\theta _{k})\,}"></span> are independent, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha _{2}\theta _{2}X_{1}}{\alpha _{1}\theta _{1}X_{2}}}\sim \mathrm {F} (2\alpha _{1},2\alpha _{2})}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mrow>
|
|||
|
|
<msub>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<msub>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<msub>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<msub>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<msub>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<msub>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>∼<!-- ∼ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi mathvariant="normal">F</mi>
|
|||
|
|
</mrow>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
<msub>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
<msub>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha _{2}\theta _{2}X_{1}}{\alpha _{1}\theta _{1}X_{2}}}\sim \mathrm {F} (2\alpha _{1},2\alpha _{2})}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e8dddf330ecfa29ac3839ed463ee66efe4e3f93" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.369ex; height:5.843ex;" alt="{\displaystyle {\frac {\alpha _{2}\theta _{2}X_{1}}{\alpha _{1}\theta _{1}X_{2}}}\sim \mathrm {F} (2\alpha _{1},2\alpha _{2})}"></span>, or equivalently, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {X_{1}}{X_{2}}}\sim \beta '\left(\alpha _{1},\alpha _{2},1,{\frac {\theta _{1}}{\theta _{2}}}\right)}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<msub>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<msub>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>∼<!-- ∼ --></mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mo>′</mo>
|
|||
|
|
</msup>
|
|||
|
|
<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<mrow>
|
|||
|
|
<msub>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<msub>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<msub>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\frac {X_{1}}{X_{2}}}\sim \beta '\left(\alpha _{1},\alpha _{2},1,{\frac {\theta _{1}}{\theta _{2}}}\right)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18ad8b86fd514386d69012b06ed5a2724f3141eb" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.072ex; height:6.176ex;" alt="{\displaystyle {\frac {X_{1}}{X_{2}}}\sim \beta '\left(\alpha _{1},\alpha _{2},1,{\frac {\theta _{1}}{\theta _{2}}}\right)}"></span></li>
|
|||
|
|
<li>Parametrization 2: If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}\sim \Gamma (\alpha _{k},\beta _{k})\,}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<msub>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>∼<!-- ∼ --></mo>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mspace width="thinmathspace" />
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle X_{k}\sim \Gamma (\alpha _{k},\beta _{k})\,}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93b3bb7f3563f635649d7dc74ad13f248e80bd53" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.775ex; height:2.843ex;" alt="{\displaystyle X_{k}\sim \Gamma (\alpha _{k},\beta _{k})\,}"></span> are independent, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha _{2}\beta _{1}X_{1}}{\alpha _{1}\beta _{2}X_{2}}}\sim \mathrm {F} (2\alpha _{1},2\alpha _{2})}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mrow>
|
|||
|
|
<msub>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<msub>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<msub>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<msub>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<msub>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<msub>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>∼<!-- ∼ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi mathvariant="normal">F</mi>
|
|||
|
|
</mrow>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
<msub>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
<msub>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha _{2}\beta _{1}X_{1}}{\alpha _{1}\beta _{2}X_{2}}}\sim \mathrm {F} (2\alpha _{1},2\alpha _{2})}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89d4abcc11916af8cb166426737f02fc6feed39e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.595ex; height:5.843ex;" alt="{\displaystyle {\frac {\alpha _{2}\beta _{1}X_{1}}{\alpha _{1}\beta _{2}X_{2}}}\sim \mathrm {F} (2\alpha _{1},2\alpha _{2})}"></span>, or equivalently, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {X_{1}}{X_{2}}}\sim \beta '\left(\alpha _{1},\alpha _{2},1,{\frac {\beta _{2}}{\beta _{1}}}\right)}">
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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<mo>∼<!-- ∼ --></mo>
|
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<msup>
|
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<mi>β<!-- β --></mi>
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<mo>′</mo>
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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<mn>2</mn>
|
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|
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|
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|
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|
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|
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|
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|
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<annotation encoding="application/x-tex">{\displaystyle {\frac {X_{1}}{X_{2}}}\sim \beta '\left(\alpha _{1},\alpha _{2},1,{\frac {\beta _{2}}{\beta _{1}}}\right)}</annotation>
|
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|
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</semantics>
|
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8feeac0cc0fe85a8a361119b77b78cf628096f3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.297ex; height:6.176ex;" alt="{\displaystyle {\frac {X_{1}}{X_{2}}}\sim \beta '\left(\alpha _{1},\alpha _{2},1,{\frac {\beta _{2}}{\beta _{1}}}\right)}"></span></li>
|
|||
|
|
<li>If <i>X</i> ~ Gamma(<i>α</i>, <i>θ</i>) and <i>Y</i> ~ Gamma(<i>β</i>, <i>θ</i>) are independently distributed, then <i>X</i>/(<i>X</i> + <i>Y</i>) has a <a href="/wiki/Beta_distribution" title="Beta distribution">beta distribution</a> with parameters <i>α</i> and <i>β</i>, and <i>X</i>/(<i>X</i> + <i>Y</i>) is independent of <i>X</i> + <i>Y</i>, which is Gamma(<i>α</i> + <i>β</i>, <i>θ</i>)-distributed.</li>
|
|||
|
|
<li>If <i>X</i><sub><i>i</i></sub> ~ Gamma(<i>α</i><sub><i>i</i></sub>, 1) are independently distributed, then the vector (<i>X</i><sub>1</sub>/<i>S</i>, ..., <i>X<sub>n</sub></i>/<i>S</i>), where <i>S</i> = <i>X</i><sub>1</sub> + ... + <i>X<sub>n</sub></i>, follows a <a href="/wiki/Dirichlet_distribution" title="Dirichlet distribution">Dirichlet distribution</a> with parameters <i>α</i><sub>1</sub>, ..., <i>α</i><sub><i>n</i></sub>.</li>
|
|||
|
|
<li>For large <i>k</i> the gamma distribution converges to <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> with mean <i>μ</i> = <i>kθ</i> and variance <i>σ</i><sup>2</sup> = <i>kθ</i><sup>2</sup>.</li>
|
|||
|
|
<li>The gamma distribution is the <a href="/wiki/Conjugate_prior" title="Conjugate prior">conjugate prior</a> for the precision of the <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> with known <a href="/wiki/Mean" title="Mean">mean</a>.</li>
|
|||
|
|
<li>The <a href="/wiki/Matrix_gamma_distribution" title="Matrix gamma distribution">matrix gamma distribution</a> and the <a href="/wiki/Wishart_distribution" title="Wishart distribution">Wishart distribution</a> are multivariate generalizations of the gamma distribution (samples are positive-definite matrices rather than positive real numbers).</li>
|
|||
|
|
<li>The gamma distribution is a special case of the <a href="/wiki/Generalized_gamma_distribution" title="Generalized gamma distribution">generalized gamma distribution</a>, the <a href="/wiki/Generalized_integer_gamma_distribution" title="Generalized integer gamma distribution">generalized integer gamma distribution</a>, and the <a href="/wiki/Generalized_inverse_Gaussian_distribution" title="Generalized inverse Gaussian distribution">generalized inverse Gaussian distribution</a>.</li>
|
|||
|
|
<li>Among the discrete distributions, the <a href="/wiki/Negative_binomial_distribution" title="Negative binomial distribution">negative binomial distribution</a> is sometimes considered the discrete analog of the gamma distribution.</li>
|
|||
|
|
<li><a href="/wiki/Tweedie_distribution" title="Tweedie distribution">Tweedie distributions</a> – the gamma distribution is a member of the family of Tweedie <a href="/wiki/Exponential_dispersion_model" title="Exponential dispersion model">exponential dispersion models</a>.</li>
|
|||
|
|
<li>Modified <a href="/wiki/Half-normal_distribution" title="Half-normal distribution">Half-normal distribution</a> – the Gamma distribution is a member of the family of <a href="/wiki/Modified_half-normal_distribution" title="Modified half-normal distribution">Modified half-normal distribution</a>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17">[17]</a></sup> The corresponding density is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x\mid \alpha ,\beta ,\gamma )={\frac {2\beta ^{\frac {\alpha }{2}}x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\Psi {\left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}}}">
|
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<semantics>
|
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|
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|
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<mo stretchy="false">(</mo>
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<mi>x</mi>
|
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<mo>∣<!-- ∣ --></mo>
|
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<mi>α<!-- α --></mi>
|
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<mo>,</mo>
|
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|
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<mo>,</mo>
|
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<mi>γ<!-- γ --></mi>
|
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<mo stretchy="false">)</mo>
|
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|
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|
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<mfrac>
|
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|
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|
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|
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</mfrac>
|
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|
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</mrow>
|
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</msup>
|
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|
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<mi>x</mi>
|
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|
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<mrow class="MJX-TeXAtom-ORD">
|
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|
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<mi>α<!-- α --></mi>
|
|||
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<mo>−<!-- − --></mo>
|
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<mn>1</mn>
|
|||
|
|
</mrow>
|
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|
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</msup>
|
|||
|
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<mi>exp</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
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|
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<mo stretchy="false">(</mo>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>β<!-- β --></mi>
|
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|
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<msup>
|
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|
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<mi>x</mi>
|
|||
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|
<mrow class="MJX-TeXAtom-ORD">
|
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|
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|
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|
|||
|
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<mo>+</mo>
|
|||
|
|
<mi>γ<!-- γ --></mi>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
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|
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|
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|
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<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
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<mrow>
|
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<mo>(</mo>
|
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|
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|
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|
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|
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|
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<mn>2</mn>
|
|||
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</mfrac>
|
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|
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<mo>,</mo>
|
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<mrow class="MJX-TeXAtom-ORD">
|
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|
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|
|||
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<msqrt>
|
|||
|
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<mi>β<!-- β --></mi>
|
|||
|
|
</msqrt>
|
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|
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|
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|
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<mo>)</mo>
|
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|
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|
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|
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|
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|
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|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle f(x\mid \alpha ,\beta ,\gamma )={\frac {2\beta ^{\frac {\alpha }{2}}x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\Psi {\left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d857ffeb61fc56ae24b19d6a4dbe58e0b8b9f4c6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:42.281ex; height:10.509ex;" alt="{\displaystyle f(x\mid \alpha ,\beta ,\gamma )={\frac {2\beta ^{\frac {\alpha }{2}}x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\Psi {\left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)}">
|
|||
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|
<semantics>
|
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|
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<mstyle displaystyle="true" scriptlevel="0">
|
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<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
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<mi>α<!-- α --></mi>
|
|||
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<mo>,</mo>
|
|||
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<mi>z</mi>
|
|||
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|
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|
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|
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|
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|
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|
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|
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<mo>(</mo>
|
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|
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|
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|
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|
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|
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|
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<mo>(</mo>
|
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|
|||
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|
|||
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<mo>,</mo>
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|||
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|
|||
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<mo>,</mo>
|
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|
|||
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|
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|
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|
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|
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|
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<mo>;</mo>
|
|||
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<mi>z</mi>
|
|||
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|
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<mo>)</mo>
|
|||
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|
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|
|||
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|
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|
|
<annotation encoding="application/x-tex">{\displaystyle \Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f318f1c6f5b6c50886d35fe09b9205c3e66784" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.494ex; height:7.509ex;" alt="{\displaystyle \Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)}"></span> denotes the <a href="/wiki/Fox%E2%80%93Wright_Psi_function" class="mw-redirect" title="Fox–Wright Psi function">Fox–Wright Psi function</a>.</li>
|
|||
|
|
<li>For the shape-scale parameterization <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x|\theta \sim \Gamma (k,\theta )}">
|
|||
|
|
<semantics>
|
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|
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<mstyle displaystyle="true" scriptlevel="0">
|
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<mi>x</mi>
|
|||
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|
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<mo stretchy="false">|</mo>
|
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|
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<mi>θ<!-- θ --></mi>
|
|||
|
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<mo>∼<!-- ∼ --></mo>
|
|||
|
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<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
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<mo stretchy="false">(</mo>
|
|||
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<mi>k</mi>
|
|||
|
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<mo>,</mo>
|
|||
|
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<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle x|\theta \sim \Gamma (k,\theta )}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0acf17a39a34862746426e99e25aa8ee005b3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.763ex; height:2.843ex;" alt="{\displaystyle x|\theta \sim \Gamma (k,\theta )}"></span>, if the scale parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \sim IG(b,1)}">
|
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|
<semantics>
|
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|
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<mstyle displaystyle="true" scriptlevel="0">
|
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<mi>θ<!-- θ --></mi>
|
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<mo>∼<!-- ∼ --></mo>
|
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<mi>I</mi>
|
|||
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<mi>G</mi>
|
|||
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<mo stretchy="false">(</mo>
|
|||
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<mi>b</mi>
|
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|
|
<mo>,</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \theta \sim IG(b,1)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf7c9a793a31d07edd3f6d59d75202965b08b09d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.191ex; height:2.843ex;" alt="{\displaystyle \theta \sim IG(b,1)}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle IG}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>I</mi>
|
|||
|
|
<mi>G</mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle IG}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1cbed103b7527b8761b26fb2b729ad65844d7a6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.998ex; height:2.176ex;" alt="{\displaystyle IG}"></span> denotes the <a href="/wiki/Inverse-gamma_distribution" title="Inverse-gamma distribution">Inverse-gamma distribution</a>, then the marginal distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\sim \beta '(k,b)}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo>∼<!-- ∼ --></mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mo>′</mo>
|
|||
|
|
</msup>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>b</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle x\sim \beta '(k,b)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a350b021b5230bc13e56f87a39b1c64f816c04c3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.502ex; height:3.009ex;" alt="{\displaystyle x\sim \beta '(k,b)}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta '}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<msup>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mo>′</mo>
|
|||
|
|
</msup>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \beta '}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d14001f211d8e272b5c10e45c739d320359c48c8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.022ex; height:2.843ex;" alt="\beta '"></span> denotes the <a href="/wiki/Beta_prime_distribution" title="Beta prime distribution">Beta prime distribution</a>.</li></ul>
|
|||
|
|
<h3><span class="mw-headline" id="Compound_gamma">Compound gamma</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=18" title="Edit section: Compound gamma"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
|
|||
|
|
<p>If the shape parameter of the gamma distribution is known, but the inverse-scale parameter is unknown, then a gamma distribution for the inverse scale forms a conjugate prior. The <a href="/wiki/Compound_distribution" class="mw-redirect" title="Compound distribution">compound distribution</a>, which results from integrating out the inverse scale, has a closed-form solution known as the <a href="/wiki/Compound_gamma_distribution" class="mw-redirect" title="Compound gamma distribution">compound gamma distribution</a>.<sup id="cite_ref-Dubey_18-0" class="reference"><a href="#cite_note-Dubey-18">[18]</a></sup>
|
|||
|
|
</p><p>If, instead, the shape parameter is known but the mean is unknown, with the prior of the mean being given by another gamma distribution, then it results in <a href="/wiki/K-distribution" title="K-distribution">K-distribution</a>.
|
|||
|
|
</p>
|
|||
|
|
<h3><span class="mw-headline" id="Weibull_and_stable_count">Weibull and stable count</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=19" title="Edit section: Weibull and stable count"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
|
|||
|
|
<p>The gamma distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x;k)\,(k>1)}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>f</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo>;</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mspace width="thinmathspace" />
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle f(x;k)\,(k>1)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6e5dddb7d9a9add51167f6c7786a39a798895f8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.331ex; height:2.843ex;" alt="{\displaystyle f(x;k)\,(k>1)}"></span> can be expressed as the product distribution of a <a href="/wiki/Weibull_distribution" title="Weibull distribution">Weibull distribution</a> and a variant form of the <a href="/wiki/Stable_count_distribution" title="Stable count distribution">stable count distribution</a>.
|
|||
|
|
Its shape parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="k"></span> can be regarded as the inverse of Lévy's stability parameter in the stable count distribution:
|
|||
|
|
<div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x;k)=\displaystyle \int _{0}^{\infty }{\frac {1}{u}}\,W_{k}\left({\frac {x}{u}}\right)\left[ku^{k-1}\,{\mathfrak {N}}_{\frac {1}{k}}\left(u^{k}\right)\right]\,du,}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>f</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo>;</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<msubsup>
|
|||
|
|
<mo>∫<!-- ∫ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>0</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi mathvariant="normal">∞<!-- ∞ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msubsup>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mi>u</mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mspace width="thinmathspace" />
|
|||
|
|
<msub>
|
|||
|
|
<mi>W</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mi>u</mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mo>[</mo>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<msup>
|
|||
|
|
<mi>u</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mspace width="thinmathspace" />
|
|||
|
|
<msub>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi mathvariant="fraktur">N</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>u</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>]</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mspace width="thinmathspace" />
|
|||
|
|
<mi>d</mi>
|
|||
|
|
<mi>u</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle f(x;k)=\displaystyle \int _{0}^{\infty }{\frac {1}{u}}\,W_{k}\left({\frac {x}{u}}\right)\left[ku^{k-1}\,{\mathfrak {N}}_{\frac {1}{k}}\left(u^{k}\right)\right]\,du,}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7fdb1229d8546f9459b5bfdae27e98ee789a192" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.21ex; height:6.176ex;" alt="{\displaystyle f(x;k)=\displaystyle \int _{0}^{\infty }{\frac {1}{u}}\,W_{k}\left({\frac {x}{u}}\right)\left[ku^{k-1}\,{\mathfrak {N}}_{\frac {1}{k}}\left(u^{k}\right)\right]\,du,}"></div>
|
|||
|
|
where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {N}}_{\alpha }(\nu )}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<msub>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi mathvariant="fraktur">N</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>ν<!-- ν --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {N}}_{\alpha }(\nu )}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16503339cce78afe1b7b86dcf6d064fb7f34b979" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.259ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {N}}_{\alpha }(\nu )}"></span> is a standard stable count distribution of shape <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =1/k}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>/</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \alpha =1/k}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfe1c09916f6155dccb18cd323dcce7170ad5fcd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.122ex; height:2.843ex;" alt="{\displaystyle \alpha =1/k}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{k}(x)}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<msub>
|
|||
|
|
<mi>W</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle W_{k}(x)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cb78f76255ce2e2a500863818ce36691169fb9e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.421ex; height:2.843ex;" alt="{\displaystyle W_{k}(x)}"></span> is a standard Weibull distribution of shape <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="k"></span>.
|
|||
|
|
</p>
|
|||
|
|
<h2><span class="mw-headline" id="Statistical_inference">Statistical inference</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=20" title="Edit section: Statistical inference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
|
|||
|
|
<h3><span class="mw-headline" id="Parameter_estimation">Parameter estimation</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=21" title="Edit section: Parameter estimation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
|
|||
|
|
<h4><span class="mw-headline" id="Maximum_likelihood_estimation">Maximum likelihood estimation</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=22" title="Edit section: Maximum likelihood estimation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h4>
|
|||
|
|
<p>The likelihood function for <i>N</i> <a href="/wiki/Independent_and_identically-distributed_random_variables" class="mw-redirect" title="Independent and identically-distributed random variables">iid</a> observations (<i>x</i><sub>1</sub>, ..., <i>x</i><sub><i>N</i></sub>) is
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(k,\theta )=\prod _{i=1}^{N}f(x_{i};k,\theta )}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
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<mstyle displaystyle="true" scriptlevel="0">
|
|||
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<mi>L</mi>
|
|||
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<mo stretchy="false">(</mo>
|
|||
|
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<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
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<mo>=</mo>
|
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<munderover>
|
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<mo>∏<!-- ∏ --></mo>
|
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<mrow class="MJX-TeXAtom-ORD">
|
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<mi>i</mi>
|
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|
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|
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</mrow>
|
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<mrow class="MJX-TeXAtom-ORD">
|
|||
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<mi>N</mi>
|
|||
|
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</mrow>
|
|||
|
|
</munderover>
|
|||
|
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<mi>f</mi>
|
|||
|
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<mo stretchy="false">(</mo>
|
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|
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<msub>
|
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|
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<mi>x</mi>
|
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<mrow class="MJX-TeXAtom-ORD">
|
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<mi>i</mi>
|
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</mrow>
|
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</msub>
|
|||
|
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<mo>;</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle L(k,\theta )=\prod _{i=1}^{N}f(x_{i};k,\theta )}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e59f233200284ea82290f9fe3d1055c1f8bc02" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.77ex; height:7.343ex;" alt="L(k, \theta) = \prod_{i=1}^N f(x_i;k,\theta)"></span></dd></dl>
|
|||
|
|
<p>from which we calculate the log-likelihood function
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell (k,\theta )=(k-1)\sum _{i=1}^{N}\ln(x_{i})-\sum _{i=1}^{N}{\frac {x_{i}}{\theta }}-Nk\ln(\theta )-N\ln(\Gamma (k))}">
|
|||
|
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<semantics>
|
|||
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<mrow class="MJX-TeXAtom-ORD">
|
|||
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<mstyle displaystyle="true" scriptlevel="0">
|
|||
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<mi>ℓ<!-- ℓ --></mi>
|
|||
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<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
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<mo>=</mo>
|
|||
|
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<mo stretchy="false">(</mo>
|
|||
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<mi>k</mi>
|
|||
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<mo>−<!-- − --></mo>
|
|||
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<mn>1</mn>
|
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<mo stretchy="false">)</mo>
|
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<munderover>
|
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<mo>∑<!-- ∑ --></mo>
|
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<mrow class="MJX-TeXAtom-ORD">
|
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<mi>i</mi>
|
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|
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|
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|
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<mrow class="MJX-TeXAtom-ORD">
|
|||
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<mi>N</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</munderover>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
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<msub>
|
|||
|
|
<mi>x</mi>
|
|||
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<mrow class="MJX-TeXAtom-ORD">
|
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|
|||
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|
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|
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<mo stretchy="false">)</mo>
|
|||
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<mo>−<!-- − --></mo>
|
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<munderover>
|
|||
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<mo>∑<!-- ∑ --></mo>
|
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|
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|
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|
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|
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</mrow>
|
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<mrow class="MJX-TeXAtom-ORD">
|
|||
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|
|||
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|
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|
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|
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<mfrac>
|
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<msub>
|
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<mi>x</mi>
|
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<mrow class="MJX-TeXAtom-ORD">
|
|||
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<mi>i</mi>
|
|||
|
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|
|||
|
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</msub>
|
|||
|
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<mi>θ<!-- θ --></mi>
|
|||
|
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</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>N</mi>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>N</mi>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \ell (k,\theta )=(k-1)\sum _{i=1}^{N}\ln(x_{i})-\sum _{i=1}^{N}{\frac {x_{i}}{\theta }}-Nk\ln(\theta )-N\ln(\Gamma (k))}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d47fa11f72cfe8b701019e1e01674b0e6c593e0b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:60.646ex; height:7.343ex;" alt="{\displaystyle \ell (k,\theta )=(k-1)\sum _{i=1}^{N}\ln(x_{i})-\sum _{i=1}^{N}{\frac {x_{i}}{\theta }}-Nk\ln(\theta )-N\ln(\Gamma (k))}"></span></dd></dl>
|
|||
|
|
<p>Finding the maximum with respect to <i>θ</i> by taking the derivative and setting it equal to zero yields the <a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">maximum likelihood</a> estimator of the <i>θ</i> parameter, which equals the <a href="/wiki/Sample_mean" class="mw-redirect" title="Sample mean">sample mean</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {x}}}">
|
|||
|
|
<semantics>
|
|||
|
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<mrow class="MJX-TeXAtom-ORD">
|
|||
|
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<mstyle displaystyle="true" scriptlevel="0">
|
|||
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<mrow class="MJX-TeXAtom-ORD">
|
|||
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<mrow class="MJX-TeXAtom-ORD">
|
|||
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<mover>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo stretchy="false">¯<!-- ¯ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\bar {x}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/466e03e1c9533b4dab1b9949dad393883f385d80" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.009ex;" alt="{\bar {x}}"></span> divided by the shape parameter <i>k</i>:
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\theta }}={\frac {1}{kN}}\sum _{i=1}^{N}x_{i}={\frac {\bar {x}}{k}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
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<mstyle displaystyle="true" scriptlevel="0">
|
|||
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|
|||
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<mrow class="MJX-TeXAtom-ORD">
|
|||
|
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<mover>
|
|||
|
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<mi>θ<!-- θ --></mi>
|
|||
|
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<mo stretchy="false">^<!-- ^ --></mo>
|
|||
|
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</mover>
|
|||
|
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|
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|
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|
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|
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|
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|
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|
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|
|||
|
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<mi>N</mi>
|
|||
|
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</mrow>
|
|||
|
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</mfrac>
|
|||
|
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</mrow>
|
|||
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<munderover>
|
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|
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|
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|
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|
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|
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|
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<mrow class="MJX-TeXAtom-ORD">
|
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|
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|
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|
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<msub>
|
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|
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|
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|
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|
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|
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|
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|
|||
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|
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|
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</mover>
|
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</mrow>
|
|||
|
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<mi>k</mi>
|
|||
|
|
</mfrac>
|
|||
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|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\hat {\theta }}={\frac {1}{kN}}\sum _{i=1}^{N}x_{i}={\frac {\bar {x}}{k}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b434bb6cc9b63502d363cd5396852735fb532cbf" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.088ex; height:7.343ex;" alt="{\displaystyle {\hat {\theta }}={\frac {1}{kN}}\sum _{i=1}^{N}x_{i}={\frac {\bar {x}}{k}}}"></span></dd></dl>
|
|||
|
|
<p>Substituting this into the log-likelihood function gives
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell (k)=(k-1)\sum _{i=1}^{N}\ln(x_{i})-Nk-Nk\ln \left({\frac {\sum x_{i}}{kN}}\right)-N\ln(\Gamma (k))}">
|
|||
|
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<semantics>
|
|||
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<mrow class="MJX-TeXAtom-ORD">
|
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<mstyle displaystyle="true" scriptlevel="0">
|
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<mi>ℓ<!-- ℓ --></mi>
|
|||
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<mo stretchy="false">(</mo>
|
|||
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<mi>k</mi>
|
|||
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|
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|
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|
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|
|||
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<mo>−<!-- − --></mo>
|
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|
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|
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|
|||
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|
|||
|
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<mi>ln</mi>
|
|||
|
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<mo>⁡<!-- --></mo>
|
|||
|
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<mo stretchy="false">(</mo>
|
|||
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|
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|
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|
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|
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<mo stretchy="false">)</mo>
|
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<mo>−<!-- − --></mo>
|
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<mi>N</mi>
|
|||
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<mi>k</mi>
|
|||
|
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<mo>−<!-- − --></mo>
|
|||
|
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<mi>N</mi>
|
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<mi>k</mi>
|
|||
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<mi>ln</mi>
|
|||
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<mo>⁡<!-- --></mo>
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|
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|
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|
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|
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|
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|
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|
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<mrow>
|
|||
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<mi>k</mi>
|
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<mi>N</mi>
|
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|
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</mfrac>
|
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|
|||
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<mo>)</mo>
|
|||
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|
|||
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<mo>−<!-- − --></mo>
|
|||
|
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<mi>N</mi>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
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|
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<mo stretchy="false">(</mo>
|
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|
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<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
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<mo stretchy="false">(</mo>
|
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<mi>k</mi>
|
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|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \ell (k)=(k-1)\sum _{i=1}^{N}\ln(x_{i})-Nk-Nk\ln \left({\frac {\sum x_{i}}{kN}}\right)-N\ln(\Gamma (k))}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dde8213005c7e787111a2390b09cd5f5372047a0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:61.417ex; height:7.343ex;" alt="{\displaystyle \ell (k)=(k-1)\sum _{i=1}^{N}\ln(x_{i})-Nk-Nk\ln \left({\frac {\sum x_{i}}{kN}}\right)-N\ln(\Gamma (k))}"></span></dd></dl>
|
|||
|
|
<p>We need at least two samples: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\geq 2}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>N</mi>
|
|||
|
|
<mo>≥<!-- ≥ --></mo>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle N\geq 2}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a366b1cf2a65d58e6be08fede1a447e0771926b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.325ex; height:2.343ex;" alt="{\displaystyle N\geq 2}"></span>, because for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=1}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>N</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle N=1}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85982022b9eb1f295b44de55023687a490db0a39" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.325ex; height:2.176ex;" alt="N=1"></span>, the function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell (k)}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>ℓ<!-- ℓ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \ell (k)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f050e906764d1b0f675a509a5a34a818f656aee0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.99ex; height:2.843ex;" alt="{\displaystyle \ell (k)}"></span> increases without bounds as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\to \infty }">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">→<!-- → --></mo>
|
|||
|
|
<mi mathvariant="normal">∞<!-- ∞ --></mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k\to \infty }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acf168f27ae911d0e1f71ce7e2c8fc9d1a3adeb5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.149ex; height:2.176ex;" alt="k\to \infty "></span>. For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>0}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>></mo>
|
|||
|
|
<mn>0</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k>0}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27b3af208b148139eefc03f0f80fa94c38c5af45" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="k>0"></span>, it can be verified that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell (k)}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>ℓ<!-- ℓ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \ell (k)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f050e906764d1b0f675a509a5a34a818f656aee0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.99ex; height:2.843ex;" alt="{\displaystyle \ell (k)}"></span> is strictly <a href="/wiki/Concave_function" title="Concave function">concave</a>, by using <a href="/wiki/Polygamma_function#Inequalities" title="Polygamma function">inequality properties of the polygamma function</a>. Finding the maximum with respect to <i>k</i> by taking the derivative and setting it equal to zero yields
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(k)-\psi (k)=\ln \left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}\right)-{\frac {1}{N}}\sum _{i=1}^{N}\ln(x_{i})=\ln({\bar {x}})-{\overline {\ln(x)}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>ψ<!-- ψ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<munderover>
|
|||
|
|
<mo>∑<!-- ∑ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</munderover>
|
|||
|
|
<msub>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<munderover>
|
|||
|
|
<mo>∑<!-- ∑ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</munderover>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo stretchy="false">¯<!-- ¯ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mo accent="false">¯<!-- ¯ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \ln(k)-\psi (k)=\ln \left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}\right)-{\frac {1}{N}}\sum _{i=1}^{N}\ln(x_{i})=\ln({\bar {x}})-{\overline {\ln(x)}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a03ed66e481ae4498fe0895347a0de878bc98783" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:62.169ex; height:7.509ex;" alt="{\displaystyle \ln(k)-\psi (k)=\ln \left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}\right)-{\frac {1}{N}}\sum _{i=1}^{N}\ln(x_{i})=\ln({\bar {x}})-{\overline {\ln(x)}}}"></span></dd></dl>
|
|||
|
|
<p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>ψ<!-- ψ --></mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="\psi "></span> is the <a href="/wiki/Digamma_function" title="Digamma function">digamma function</a> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\ln(x)}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mo accent="false">¯<!-- ¯ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\overline {\ln(x)}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad970f26e684357434e66c18fb225775bab8d99c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.193ex; height:3.676ex;" alt="{\displaystyle {\overline {\ln(x)}}}"></span> is the sample mean of ln(<i>x</i>). There is no closed-form solution for <i>k</i>. The function is numerically very well behaved, so if a numerical solution is desired, it can be found using, for example, <a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a>. An initial value of <i>k</i> can be found either using the <a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">method of moments</a>, or using the approximation
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(k)-\psi (k)\approx {\frac {1}{2k}}\left(1+{\frac {1}{6k+1}}\right)}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>ψ<!-- ψ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>≈<!-- ≈ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mrow>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<mrow>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mrow>
|
|||
|
|
<mn>6</mn>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \ln(k)-\psi (k)\approx {\frac {1}{2k}}\left(1+{\frac {1}{6k+1}}\right)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc17e7e8fea23d287a45596501348a1bd573108" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.666ex; height:6.176ex;" alt="\ln(k) - \psi(k) \approx \frac{1}{2k}\left(1 + \frac{1}{6k + 1}\right)"></span></dd></dl>
|
|||
|
|
<p>If we let
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\ln \left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}\right)-{\frac {1}{N}}\sum _{i=1}^{N}\ln(x_{i})=\ln({\bar {x}})-{\overline {\ln(x)}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>s</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
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|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<munderover>
|
|||
|
|
<mo>∑<!-- ∑ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</munderover>
|
|||
|
|
<msub>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<munderover>
|
|||
|
|
<mo>∑<!-- ∑ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</munderover>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo stretchy="false">¯<!-- ¯ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mo accent="false">¯<!-- ¯ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle s=\ln \left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}\right)-{\frac {1}{N}}\sum _{i=1}^{N}\ln(x_{i})=\ln({\bar {x}})-{\overline {\ln(x)}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87a68086a436ade48042682041f017437611272" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:50.926ex; height:7.509ex;" alt="{\displaystyle s=\ln \left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}\right)-{\frac {1}{N}}\sum _{i=1}^{N}\ln(x_{i})=\ln({\bar {x}})-{\overline {\ln(x)}}}"></span></dd></dl>
|
|||
|
|
<p>then <i>k</i> is approximately
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\approx {\frac {3-s+{\sqrt {(s-3)^{2}+24s}}}{12s}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>≈<!-- ≈ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mrow>
|
|||
|
|
<mn>3</mn>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>s</mi>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<msqrt>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>s</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>3</mn>
|
|||
|
|
<msup>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mn>24</mn>
|
|||
|
|
<mi>s</mi>
|
|||
|
|
</msqrt>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mn>12</mn>
|
|||
|
|
<mi>s</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k\approx {\frac {3-s+{\sqrt {(s-3)^{2}+24s}}}{12s}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb19579e029ef52f2677c6aaa01d443a93dc4a85" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.616ex; height:6.176ex;" alt="k \approx \frac{3 - s + \sqrt{(s - 3)^2 + 24s}}{12s}"></span></dd></dl>
|
|||
|
|
<p>which is within 1.5% of the correct value.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19">[19]</a></sup> An explicit form for the Newton–Raphson update of this initial guess is:<sup id="cite_ref-20" class="reference"><a href="#cite_note-20">[20]</a></sup>
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\leftarrow k-{\frac {\ln(k)-\psi (k)-s}{{\frac {1}{k}}-\psi ^{\prime }(k)}}.}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">←<!-- ← --></mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>ψ<!-- ψ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>s</mi>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>ψ<!-- ψ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>.</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k\leftarrow k-{\frac {\ln(k)-\psi (k)-s}{{\frac {1}{k}}-\psi ^{\prime }(k)}}.}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0096669070e515dcfa6105c0c1bb4785925cd765" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:26.625ex; height:7.343ex;" alt="{\displaystyle k\leftarrow k-{\frac {\ln(k)-\psi (k)-s}{{\frac {1}{k}}-\psi ^{\prime }(k)}}.}"></span></dd></dl>
|
|||
|
|
<p>At the maximum-likelihood estimate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\hat {k}},{\hat {\theta }})}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">^<!-- ^ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">^<!-- ^ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle ({\hat {k}},{\hat {\theta }})}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77b7912fe7770f431c4e61a0e4595918a0848e49" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.41ex; height:3.343ex;" alt="{\displaystyle ({\hat {k}},{\hat {\theta }})}"></span>, the expected values for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>x</mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="x"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(x)}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \ln(x)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0df055b8e294310e6785701c1c67105e109191d8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.078ex; height:2.843ex;" alt="\ln(x)"></span> agree with the empirical averages:
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\hat {k}}{\hat {\theta }}&={\bar {x}}&&{\text{and}}&\psi ({\hat {k}})+\ln({\hat {\theta }})&={\overline {\ln(x)}}.\end{aligned}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
|
|||
|
|
<mtr>
|
|||
|
|
<mtd>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">^<!-- ^ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">^<!-- ^ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
</mtd>
|
|||
|
|
<mtd>
|
|||
|
|
<mi></mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo stretchy="false">¯<!-- ¯ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
</mtd>
|
|||
|
|
<mtd />
|
|||
|
|
<mtd>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mtext>and</mtext>
|
|||
|
|
</mrow>
|
|||
|
|
</mtd>
|
|||
|
|
<mtd>
|
|||
|
|
<mi>ψ<!-- ψ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">^<!-- ^ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">^<!-- ^ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mtd>
|
|||
|
|
<mtd>
|
|||
|
|
<mi></mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mo accent="false">¯<!-- ¯ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>.</mo>
|
|||
|
|
</mtd>
|
|||
|
|
</mtr>
|
|||
|
|
</mtable>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\hat {k}}{\hat {\theta }}&={\bar {x}}&&{\text{and}}&\psi ({\hat {k}})+\ln({\hat {\theta }})&={\overline {\ln(x)}}.\end{aligned}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5402f97eface3b69404fbf5a21f489c09eb7a543" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:42.202ex; height:3.509ex;" alt="{\displaystyle {\begin{aligned}{\hat {k}}{\hat {\theta }}&={\bar {x}}&&{\text{and}}&\psi ({\hat {k}})+\ln({\hat {\theta }})&={\overline {\ln(x)}}.\end{aligned}}}"></span></dd></dl>
|
|||
|
|
<h5><span class="mw-headline" id="Caveat_for_small_shape_parameter">Caveat for small shape parameter</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=23" title="Edit section: Caveat for small shape parameter"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h5>
|
|||
|
|
<p>For data, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},\ldots ,x_{N})}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mo>…<!-- … --></mo>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle (x_{1},\ldots ,x_{N})}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfe46c866379cec130e31f750cf08fa3ce5edfaa" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.393ex; height:2.843ex;" alt="{\displaystyle (x_{1},\ldots ,x_{N})}"></span>, that is represented in a <a href="/wiki/Floating_point" class="mw-redirect" title="Floating point">floating point</a> format that underflows to 0 for values smaller than <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>ε<!-- ε --></mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="\varepsilon "></span>, the logarithms that are needed for the maximum-likelihood estimate will cause failure if there are any underflows. If we assume the data was generated by a gamma distribution with cdf <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x;k,\theta )}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>F</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo>;</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle F(x;k,\theta )}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d52dc64c0a15754ca6e07f1772061da406e13308" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.249ex; height:2.843ex;" alt="{\displaystyle F(x;k,\theta )}"></span>, then the probability that there is at least one underflow is:
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P({\text{underflow}})=1-(1-F(\varepsilon ;k,\theta ))^{N}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>P</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mtext>underflow</mtext>
|
|||
|
|
</mrow>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>F</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>ε<!-- ε --></mi>
|
|||
|
|
<mo>;</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<msup>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle P({\text{underflow}})=1-(1-F(\varepsilon ;k,\theta ))^{N}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6de6fb21a1e6a5e2df503c4a69f91db17c876706" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.184ex; height:3.176ex;" alt="{\displaystyle P({\text{underflow}})=1-(1-F(\varepsilon ;k,\theta ))^{N}}"></span></dd></dl>
|
|||
|
|
<p>This probability will approach 1 for small <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="k"></span> and large <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle N}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="N"></span>. For example, at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=10^{-2}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<msup>
|
|||
|
|
<mn>10</mn>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k=10^{-2}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed55da0cb5f3e282ffc0d2c211ad2e7c2e24d978" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.967ex; height:2.676ex;" alt="{\displaystyle k=10^{-2}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=10^{4}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>N</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<msup>
|
|||
|
|
<mn>10</mn>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>4</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle N=10^{4}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92c893655aade068aa13c22966cb222d4bbac837" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.541ex; height:2.676ex;" alt="{\displaystyle N=10^{4}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon =2.25\times 10^{-308}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>ε<!-- ε --></mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>2.25</mn>
|
|||
|
|
<mo>×<!-- × --></mo>
|
|||
|
|
<msup>
|
|||
|
|
<mn>10</mn>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>308</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \varepsilon =2.25\times 10^{-308}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/753624cf1c18b917bc51c09f9bc7261d1b1fbbe8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.458ex; height:2.676ex;" alt="{\displaystyle \varepsilon =2.25\times 10^{-308}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P({\text{underflow}})\approx 0.9998}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>P</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mtext>underflow</mtext>
|
|||
|
|
</mrow>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>≈<!-- ≈ --></mo>
|
|||
|
|
<mn>0.9998</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle P({\text{underflow}})\approx 0.9998}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4145cd4e3821fab09005f66981c9745abd2ed752" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.133ex; height:2.843ex;" alt="{\displaystyle P({\text{underflow}})\approx 0.9998}"></span>. A workaround is to instead have the data in logarithmic format.
|
|||
|
|
</p><p>In order to test an implementation of a maximum-likelihood estimator that takes logarithmic data as input, it is useful to be able to generate non-underflowing logarithms of random gamma variates, when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k<1}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo><</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle k<1}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b098ac734780e67edb80b5a1039aea80f81595a1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k<1}"></span>. Following the implementation in <code>scipy.stats.loggamma</code>, this can be done as follows:<sup id="cite_ref-Marsaglia2000_21-0" class="reference"><a href="#cite_note-Marsaglia2000-21">[21]</a></sup> sample <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\sim {\text{Gamma}}(k+1,\theta )}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>Y</mi>
|
|||
|
|
<mo>∼<!-- ∼ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mtext>Gamma</mtext>
|
|||
|
|
</mrow>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle Y\sim {\text{Gamma}}(k+1,\theta )}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a7e4951dd26ab192dfe5719aac257ae932942a1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.041ex; height:2.843ex;" alt="{\displaystyle Y\sim {\text{Gamma}}(k+1,\theta )}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\sim {\text{Uniform}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>U</mi>
|
|||
|
|
<mo>∼<!-- ∼ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mtext>Uniform</mtext>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle U\sim {\text{Uniform}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce2d79a68d9de22a25a154cd29effa606ea01a52" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.285ex; height:2.176ex;" alt="{\displaystyle U\sim {\text{Uniform}}}"></span> independently. Then the required logarithmic sample is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=\ln(Y)+\ln(U)/k}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>Z</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>Y</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>U</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>/</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle Z=\ln(Y)+\ln(U)/k}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a8e3d978cd6908a457738820b19f2bb85e42929" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.046ex; height:2.843ex;" alt="{\displaystyle Z=\ln(Y)+\ln(U)/k}"></span>, so that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(Z)\sim {\text{Gamma}}(k,\theta )}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>exp</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>Z</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>∼<!-- ∼ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mtext>Gamma</mtext>
|
|||
|
|
</mrow>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \exp(Z)\sim {\text{Gamma}}(k,\theta )}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad42d845708982e924f81ef9a668a8c1464ba237" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.307ex; height:2.843ex;" alt="{\displaystyle \exp(Z)\sim {\text{Gamma}}(k,\theta )}"></span>.
|
|||
|
|
</p>
|
|||
|
|
<h4><span class="mw-headline" id="Closed-form_estimators">Closed-form estimators</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=24" title="Edit section: Closed-form estimators"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h4>
|
|||
|
|
<p>Consistent closed-form estimators of <i>k</i> and <i>θ</i> exists that are derived from the likelihood of the <a href="/wiki/Generalized_gamma_distribution" title="Generalized gamma distribution">generalized gamma distribution</a>.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22">[22]</a></sup>
|
|||
|
|
</p><p>The estimate for the shape <i>k</i> is
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {k}}={\frac {N\sum _{i=1}^{N}x_{i}}{N\sum _{i=1}^{N}x_{i}\ln(x_{i})-\sum _{i=1}^{N}x_{i}\sum _{i=1}^{N}\ln(x_{i})}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">^<!-- ^ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>N</mi>
|
|||
|
|
<munderover>
|
|||
|
|
<mo>∑<!-- ∑ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</munderover>
|
|||
|
|
<msub>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>N</mi>
|
|||
|
|
<munderover>
|
|||
|
|
<mo>∑<!-- ∑ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</munderover>
|
|||
|
|
<msub>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<munderover>
|
|||
|
|
<mo>∑<!-- ∑ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</munderover>
|
|||
|
|
<msub>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<munderover>
|
|||
|
|
<mo>∑<!-- ∑ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</munderover>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\hat {k}}={\frac {N\sum _{i=1}^{N}x_{i}}{N\sum _{i=1}^{N}x_{i}\ln(x_{i})-\sum _{i=1}^{N}x_{i}\sum _{i=1}^{N}\ln(x_{i})}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40265268c81fa7c8a038d63097dda9f5c62d5fb6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:44.449ex; height:7.509ex;" alt="{\displaystyle {\hat {k}}={\frac {N\sum _{i=1}^{N}x_{i}}{N\sum _{i=1}^{N}x_{i}\ln(x_{i})-\sum _{i=1}^{N}x_{i}\sum _{i=1}^{N}\ln(x_{i})}}}"></span></dd></dl>
|
|||
|
|
<p>and the estimate for the scale <i>θ</i> is
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\theta }}={\frac {1}{N^{2}}}\left(N\sum _{i=1}^{N}x_{i}\ln(x_{i})-\sum _{i=1}^{N}x_{i}\sum _{i=1}^{N}\ln(x_{i})\right)}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">^<!-- ^ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<msup>
|
|||
|
|
<mi>N</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>N</mi>
|
|||
|
|
<munderover>
|
|||
|
|
<mo>∑<!-- ∑ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</munderover>
|
|||
|
|
<msub>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<munderover>
|
|||
|
|
<mo>∑<!-- ∑ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</munderover>
|
|||
|
|
<msub>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<munderover>
|
|||
|
|
<mo>∑<!-- ∑ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</munderover>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\hat {\theta }}={\frac {1}{N^{2}}}\left(N\sum _{i=1}^{N}x_{i}\ln(x_{i})-\sum _{i=1}^{N}x_{i}\sum _{i=1}^{N}\ln(x_{i})\right)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e2c82bf844e30709858be5c870c48db24a11148" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.842ex; height:7.509ex;" alt="{\displaystyle {\hat {\theta }}={\frac {1}{N^{2}}}\left(N\sum _{i=1}^{N}x_{i}\ln(x_{i})-\sum _{i=1}^{N}x_{i}\sum _{i=1}^{N}\ln(x_{i})\right)}"></span></dd></dl>
|
|||
|
|
<p>Using the sample mean of <i>x</i>, the sample mean of ln(<i>x</i>), and the sample mean of the product <i>x</i>·ln(<i>x</i>) simplifies the expressions to:
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {k}}={\bar {x}}/{\hat {\theta }}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">^<!-- ^ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo stretchy="false">¯<!-- ¯ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>/</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">^<!-- ^ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\hat {k}}={\bar {x}}/{\hat {\theta }}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65f9ada852433174f5c66beb07973e6809458a4c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.158ex; height:3.343ex;" alt="{\displaystyle {\hat {k}}={\bar {x}}/{\hat {\theta }}}"></span></dd>
|
|||
|
|
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\theta }}={\overline {x\ln {x}}}-{\bar {x}}{\overline {\ln {x}}}.}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">^<!-- ^ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>x</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo accent="false">¯<!-- ¯ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo stretchy="false">¯<!-- ¯ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>x</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo accent="false">¯<!-- ¯ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>.</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\hat {\theta }}={\overline {x\ln {x}}}-{\bar {x}}{\overline {\ln {x}}}.}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1993cfa1e3a5633d096f41763a1fb4a2dd588b3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.53ex; height:3.176ex;" alt="{\displaystyle {\hat {\theta }}={\overline {x\ln {x}}}-{\bar {x}}{\overline {\ln {x}}}.}"></span></dd></dl>
|
|||
|
|
<p>If the rate parameterization is used, the estimate of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\beta }}=1/{\hat {\theta }}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
<mo stretchy="false">^<!-- ^ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>/</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">^<!-- ^ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\hat {\beta }}=1/{\hat {\theta }}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66836da08bc157ab0a5edf7974395e117eaee26c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.231ex; height:3.343ex;" alt="{\displaystyle {\hat {\beta }}=1/{\hat {\theta }}}"></span>.
|
|||
|
|
</p><p>These estimators are not strictly maximum likelihood estimators, but are instead referred to as mixed type log-moment estimators. They have however similar efficiency as the maximum likelihood estimators.
|
|||
|
|
</p><p>Although these estimators are consistent, they have a small bias. A bias-corrected variant of the estimator for the scale <i>θ</i> is
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\theta }}={\frac {N}{N-1}}{\hat {\theta }}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">~<!-- ~ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mi>N</mi>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>N</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">^<!-- ^ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\tilde {\theta }}={\frac {N}{N-1}}{\hat {\theta }}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69e944316bfe95be86b91886fd1ed2c430835ccd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.713ex; height:5.343ex;" alt="{\displaystyle {\tilde {\theta }}={\frac {N}{N-1}}{\hat {\theta }}}"></span></dd></dl>
|
|||
|
|
<p>A bias correction for the shape parameter <i>k</i> is given as<sup id="cite_ref-23" class="reference"><a href="#cite_note-23">[23]</a></sup>
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {k}}={\hat {k}}-{\frac {1}{N}}\left(3{\hat {k}}-{\frac {2}{3}}\left({\frac {\hat {k}}{1+{\hat {k}}}}\right)-{\frac {4}{5}}{\frac {\hat {k}}{(1+{\hat {k}})^{2}}}\right)}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">~<!-- ~ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">^<!-- ^ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<mrow>
|
|||
|
|
<mn>3</mn>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">^<!-- ^ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
<mn>3</mn>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">^<!-- ^ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">^<!-- ^ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>4</mn>
|
|||
|
|
<mn>5</mn>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">^<!-- ^ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mover>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">^<!-- ^ --></mo>
|
|||
|
|
</mover>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<msup>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\tilde {k}}={\hat {k}}-{\frac {1}{N}}\left(3{\hat {k}}-{\frac {2}{3}}\left({\frac {\hat {k}}{1+{\hat {k}}}}\right)-{\frac {4}{5}}{\frac {\hat {k}}{(1+{\hat {k}})^{2}}}\right)}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a767a0e0a4372c8ae599487d3bcb2af62413fa8f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.413ex; height:7.509ex;" alt="{\displaystyle {\tilde {k}}={\hat {k}}-{\frac {1}{N}}\left(3{\hat {k}}-{\frac {2}{3}}\left({\frac {\hat {k}}{1+{\hat {k}}}}\right)-{\frac {4}{5}}{\frac {\hat {k}}{(1+{\hat {k}})^{2}}}\right)}"></span></dd></dl>
|
|||
|
|
<h4><span class="mw-headline" id="Bayesian_minimum_mean_squared_error">Bayesian minimum mean squared error</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=25" title="Edit section: Bayesian minimum mean squared error"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h4>
|
|||
|
|
<p>With known <i>k</i> and unknown <i>θ</i>, the posterior density function for theta (using the standard scale-invariant <a href="/wiki/Prior_probability" title="Prior probability">prior</a> for <i>θ</i>) is
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(\theta \mid k,x_{1},\dots ,x_{N})\propto {\frac {1}{\theta }}\prod _{i=1}^{N}f(x_{i};k,\theta )}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>P</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo>∣<!-- ∣ --></mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mo>…<!-- … --></mo>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>∝<!-- ∝ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<munderover>
|
|||
|
|
<mo>∏<!-- ∏ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</munderover>
|
|||
|
|
<mi>f</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>;</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle P(\theta \mid k,x_{1},\dots ,x_{N})\propto {\frac {1}{\theta }}\prod _{i=1}^{N}f(x_{i};k,\theta )}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e88b49f3b39c1e9dfa4e3f11c9576df51513012d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.838ex; height:7.343ex;" alt="{\displaystyle P(\theta \mid k,x_{1},\dots ,x_{N})\propto {\frac {1}{\theta }}\prod _{i=1}^{N}f(x_{i};k,\theta )}"></span></dd></dl>
|
|||
|
|
<p>Denoting
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\equiv \sum _{i=1}^{N}x_{i},\qquad P(\theta \mid k,x_{1},\dots ,x_{N})=C(x_{i})\theta ^{-Nk-1}e^{-y/\theta }}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>y</mi>
|
|||
|
|
<mo>≡<!-- ≡ --></mo>
|
|||
|
|
<munderover>
|
|||
|
|
<mo>∑<!-- ∑ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</munderover>
|
|||
|
|
<msub>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mspace width="2em" />
|
|||
|
|
<mi>P</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo>∣<!-- ∣ --></mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mo>…<!-- … --></mo>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>N</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>C</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>N</mi>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<msup>
|
|||
|
|
<mi>e</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>y</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>/</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle y\equiv \sum _{i=1}^{N}x_{i},\qquad P(\theta \mid k,x_{1},\dots ,x_{N})=C(x_{i})\theta ^{-Nk-1}e^{-y/\theta }}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffae7a36c6134471ab6b856eb7354a68f2ee8130" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:56.041ex; height:7.343ex;" alt="{\displaystyle y\equiv \sum _{i=1}^{N}x_{i},\qquad P(\theta \mid k,x_{1},\dots ,x_{N})=C(x_{i})\theta ^{-Nk-1}e^{-y/\theta }}"></span></dd></dl>
|
|||
|
|
<p>Integration with respect to <i>θ</i> can be carried out using a change of variables, revealing that 1/<i>θ</i> is gamma-distributed with parameters <i>α</i> = <i>Nk</i>, <i>β</i> = <i>y</i>.
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{\infty }\theta ^{-Nk-1+m}e^{-y/\theta }\,d\theta =\int _{0}^{\infty }x^{Nk-1-m}e^{-xy}\,dx=y^{-(Nk-m)}\Gamma (Nk-m)\!}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<msubsup>
|
|||
|
|
<mo>∫<!-- ∫ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>0</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi mathvariant="normal">∞<!-- ∞ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msubsup>
|
|||
|
|
<msup>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>N</mi>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mi>m</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<msup>
|
|||
|
|
<mi>e</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>y</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>/</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mspace width="thinmathspace" />
|
|||
|
|
<mi>d</mi>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<msubsup>
|
|||
|
|
<mo>∫<!-- ∫ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>0</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi mathvariant="normal">∞<!-- ∞ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msubsup>
|
|||
|
|
<msup>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>N</mi>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>m</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<msup>
|
|||
|
|
<mi>e</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mi>y</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mspace width="thinmathspace" />
|
|||
|
|
<mi>d</mi>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>y</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>N</mi>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>m</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>N</mi>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>m</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mspace width="negativethinmathspace" />
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }\theta ^{-Nk-1+m}e^{-y/\theta }\,d\theta =\int _{0}^{\infty }x^{Nk-1-m}e^{-xy}\,dx=y^{-(Nk-m)}\Gamma (Nk-m)\!}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99fbb69f9a62c02e04870f6859733934f0d6c087" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-right: -0.166ex; width:69.255ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{\infty }\theta ^{-Nk-1+m}e^{-y/\theta }\,d\theta =\int _{0}^{\infty }x^{Nk-1-m}e^{-xy}\,dx=y^{-(Nk-m)}\Gamma (Nk-m)\!}"></span></dd></dl>
|
|||
|
|
<p>The moments can be computed by taking the ratio (<i>m</i> by <i>m</i> = 0)
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [x^{m}]={\frac {\Gamma (Nk-m)}{\Gamma (Nk)}}y^{m}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi mathvariant="normal">E</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">[</mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>m</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo stretchy="false">]</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mrow>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>N</mi>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>m</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>N</mi>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<msup>
|
|||
|
|
<mi>y</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>m</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [x^{m}]={\frac {\Gamma (Nk-m)}{\Gamma (Nk)}}y^{m}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61ae01ae77aa6c640cbaa1bb2a8863454827916a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.069ex; height:6.509ex;" alt="{\displaystyle \operatorname {E} [x^{m}]={\frac {\Gamma (Nk-m)}{\Gamma (Nk)}}y^{m}}"></span></dd></dl>
|
|||
|
|
<p>which shows that the mean ± standard deviation estimate of the posterior distribution for <i>θ</i> is
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {y}{Nk-1}}\pm {\sqrt {\frac {y^{2}}{(Nk-1)^{2}(Nk-2)}}}.}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mi>y</mi>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>N</mi>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>±<!-- ± --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<msqrt>
|
|||
|
|
<mfrac>
|
|||
|
|
<msup>
|
|||
|
|
<mi>y</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mrow>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>N</mi>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<msup>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>N</mi>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</msqrt>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>.</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\frac {y}{Nk-1}}\pm {\sqrt {\frac {y^{2}}{(Nk-1)^{2}(Nk-2)}}}.}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de03eb3d598f8c2c6481e9c44542904ee3858e9e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:33.989ex; height:7.676ex;" alt="{\displaystyle {\frac {y}{Nk-1}}\pm {\sqrt {\frac {y^{2}}{(Nk-1)^{2}(Nk-2)}}}.}"></span></dd></dl>
|
|||
|
|
<h3><span class="mw-headline" id="Bayesian_inference">Bayesian inference</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=26" title="Edit section: Bayesian inference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
|
|||
|
|
<h4><span class="mw-headline" id="Conjugate_prior">Conjugate prior</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=27" title="Edit section: Conjugate prior"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h4>
|
|||
|
|
<p>In <a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a>, the <b>gamma distribution</b> is the <a href="/wiki/Conjugate_prior" title="Conjugate prior">conjugate prior</a> to many likelihood distributions: the <a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson</a>, <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential</a>, <a href="/wiki/Normal_distribution" title="Normal distribution">normal</a> (with known mean), <a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto</a>, gamma with known shape <i>σ</i>, <a href="/wiki/Inverse-gamma_distribution" title="Inverse-gamma distribution">inverse gamma</a> with known shape parameter, and <a href="/wiki/Gompertz_distribution" title="Gompertz distribution">Gompertz</a> with known scale parameter.
|
|||
|
|
</p><p>The gamma distribution's <a href="/wiki/Conjugate_prior" title="Conjugate prior">conjugate prior</a> is:<sup id="cite_ref-fink_24-0" class="reference"><a href="#cite_note-fink-24">[24]</a></sup>
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(k,\theta \mid p,q,r,s)={\frac {1}{Z}}{\frac {p^{k-1}e^{-\theta ^{-1}q}}{\Gamma (k)^{r}\theta ^{ks}}},}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>p</mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo>∣<!-- ∣ --></mo>
|
|||
|
|
<mi>p</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>q</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>r</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>s</mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mi>Z</mi>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mrow>
|
|||
|
|
<msup>
|
|||
|
|
<mi>p</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<msup>
|
|||
|
|
<mi>e</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mi>q</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<msup>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>r</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<msup>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mi>s</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mrow>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle p(k,\theta \mid p,q,r,s)={\frac {1}{Z}}{\frac {p^{k-1}e^{-\theta ^{-1}q}}{\Gamma (k)^{r}\theta ^{ks}}},}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d513935bdc7db0e23f116aab5397368604572ccd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-left: -0.089ex; width:33.269ex; height:7.009ex;" alt="{\displaystyle p(k,\theta \mid p,q,r,s)={\frac {1}{Z}}{\frac {p^{k-1}e^{-\theta ^{-1}q}}{\Gamma (k)^{r}\theta ^{ks}}},}"></span></dd></dl>
|
|||
|
|
<p>where <i>Z</i> is the normalizing constant with no closed-form solution.
|
|||
|
|
The posterior distribution can be found by updating the parameters as follows:
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p'&=p\prod \nolimits _{i}x_{i},\\q'&=q+\sum \nolimits _{i}x_{i},\\r'&=r+n,\\s'&=s+n,\end{aligned}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
|
|||
|
|
<mtr>
|
|||
|
|
<mtd>
|
|||
|
|
<msup>
|
|||
|
|
<mi>p</mi>
|
|||
|
|
<mo>′</mo>
|
|||
|
|
</msup>
|
|||
|
|
</mtd>
|
|||
|
|
<mtd>
|
|||
|
|
<mi></mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>p</mi>
|
|||
|
|
<msub>
|
|||
|
|
<mo movablelimits="false">∏<!-- ∏ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<msub>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
</mtd>
|
|||
|
|
</mtr>
|
|||
|
|
<mtr>
|
|||
|
|
<mtd>
|
|||
|
|
<msup>
|
|||
|
|
<mi>q</mi>
|
|||
|
|
<mo>′</mo>
|
|||
|
|
</msup>
|
|||
|
|
</mtd>
|
|||
|
|
<mtd>
|
|||
|
|
<mi></mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>q</mi>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mo movablelimits="false">∑<!-- ∑ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<msub>
|
|||
|
|
<mi>x</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
</mtd>
|
|||
|
|
</mtr>
|
|||
|
|
<mtr>
|
|||
|
|
<mtd>
|
|||
|
|
<msup>
|
|||
|
|
<mi>r</mi>
|
|||
|
|
<mo>′</mo>
|
|||
|
|
</msup>
|
|||
|
|
</mtd>
|
|||
|
|
<mtd>
|
|||
|
|
<mi></mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>r</mi>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mi>n</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
</mtd>
|
|||
|
|
</mtr>
|
|||
|
|
<mtr>
|
|||
|
|
<mtd>
|
|||
|
|
<msup>
|
|||
|
|
<mi>s</mi>
|
|||
|
|
<mo>′</mo>
|
|||
|
|
</msup>
|
|||
|
|
</mtd>
|
|||
|
|
<mtd>
|
|||
|
|
<mi></mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>s</mi>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mi>n</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
</mtd>
|
|||
|
|
</mtr>
|
|||
|
|
</mtable>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p'&=p\prod \nolimits _{i}x_{i},\\q'&=q+\sum \nolimits _{i}x_{i},\\r'&=r+n,\\s'&=s+n,\end{aligned}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/404516eb571949b6c74db5cc102e9e5341a9de21" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:16.932ex; height:14.009ex;" alt="\begin{align} p' &= p\prod\nolimits_i x_i,\\ q' &= q + \sum\nolimits_i x_i,\\ r' &= r + n,\\ s' &= s + n, \end{align}"></span></dd></dl>
|
|||
|
|
<p>where <i>n</i> is the number of observations, and <i>x<sub>i</sub></i> is the <i>i</i>th observation.
|
|||
|
|
</p>
|
|||
|
|
<h2><span class="mw-headline" id="Occurrence_and_applications">Occurrence and applications</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=28" title="Edit section: Occurrence and applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
|
|||
|
|
<p>Consider a sequence of events, with the waiting time for each event being an exponential distribution with rate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>β<!-- β --></mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="\beta "></span>. Then the waiting time for the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
|
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|
|||
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|
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|
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|
|||
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|
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|
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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
|
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|
|||
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="n"></span>-th event to occur is the gamma distribution with integer shape <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =n}">
|
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|
|||
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|
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|
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|
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|
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|
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|
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|
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<annotation encoding="application/x-tex">{\displaystyle \alpha =n}</annotation>
|
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|
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9961376aecd3518f5e3b88e2b02034dc36ab8e4f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.981ex; height:1.676ex;" alt="{\displaystyle \alpha =n}"></span>. This construction of the gamma distribution allows it to model a wide variety of phenomena where several sub-events, each taking time with exponential distribution, must happen in sequence for a major event to occur.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25">[25]</a></sup> Examples include the waiting time of <a href="/wiki/Cell_division" title="Cell division">cell-division events</a>,<sup id="cite_ref-26" class="reference"><a href="#cite_note-26">[26]</a></sup> number of compensatory mutations for a given mutation,<sup id="cite_ref-27" class="reference"><a href="#cite_note-27">[27]</a></sup> waiting time until a repair is necessary for a hydraulic system,<sup id="cite_ref-28" class="reference"><a href="#cite_note-28">[28]</a></sup> and so on.
|
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</p><p>In biophysics, the dwell time between steps of a molecular motor like <a href="/wiki/ATP_synthase" title="ATP synthase">ATP synthase</a> is nearly exponential at constant ATP concentration, revealing that each step of the motor takes a single ATP hydrolysis. If there were n ATP hydrolysis events, then it would be a gamma distribution with degree n.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29">[29]</a></sup>
|
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</p><p>The gamma distribution has been used to model the size of <a href="/wiki/Insurance_policy" title="Insurance policy">insurance claims</a><sup id="cite_ref-30" class="reference"><a href="#cite_note-30">[30]</a></sup> and rainfalls.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31">[31]</a></sup> This means that aggregate insurance claims and the amount of rainfall accumulated in a reservoir are modelled by a <a href="/wiki/Gamma_process" title="Gamma process">gamma process</a> – much like the <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a> generates a <a href="/wiki/Poisson_process" class="mw-redirect" title="Poisson process">Poisson process</a>.
|
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</p><p>The gamma distribution is also used to model errors in multi-level <a href="/wiki/Poisson_regression" title="Poisson regression">Poisson regression</a> models because a <a href="/wiki/Mixture_distribution" title="Mixture distribution">mixture</a> of <a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distributions</a> with gamma-distributed rates has a known closed form distribution, called <a href="/wiki/Negative_binomial_distribution" title="Negative binomial distribution">negative binomial</a>.
|
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|
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</p><p>In wireless communication, the gamma distribution is used to model the <a href="/wiki/Multi-path_fading" class="mw-redirect" title="Multi-path fading">multi-path fading</a> of signal power;<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (May 2019)">citation needed</span></a></i>]</sup> see also <a href="/wiki/Rayleigh_distribution" title="Rayleigh distribution">Rayleigh distribution</a> and <a href="/wiki/Rician_distribution" class="mw-redirect" title="Rician distribution">Rician distribution</a>.
|
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</p><p>In <a href="/wiki/Oncology" title="Oncology">oncology</a>, the age distribution of <a href="/wiki/Cancer" title="Cancer">cancer</a> <a href="/wiki/Disease_incidence" class="mw-redirect" title="Disease incidence">incidence</a> often follows the gamma distribution, wherein the shape and scale parameters predict, respectively, the number of <a href="/wiki/Carcinogenesis" title="Carcinogenesis">driver events</a> and the time interval between them.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32">[32]</a></sup><sup id="cite_ref-33" class="reference"><a href="#cite_note-33">[33]</a></sup>
|
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</p><p>In <a href="/wiki/Neuroscience" title="Neuroscience">neuroscience</a>, the gamma distribution is often used to describe the distribution of <a href="/wiki/Temporal_coding" class="mw-redirect" title="Temporal coding">inter-spike intervals</a>.<sup id="cite_ref-Robson_34-0" class="reference"><a href="#cite_note-Robson-34">[34]</a></sup><sup id="cite_ref-Wright,_2015_35-0" class="reference"><a href="#cite_note-Wright,_2015-35">[35]</a></sup>
|
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</p><p>In <a href="/wiki/Bacterial_genetics" title="Bacterial genetics">bacterial</a> <a href="/wiki/Gene_expression" title="Gene expression">gene expression</a>, the <a href="/wiki/Copy_number_analysis" title="Copy number analysis">copy number</a> of a <a href="/wiki/Constitutively_expressed" class="mw-redirect" title="Constitutively expressed">constitutively expressed</a> protein often follows the gamma distribution, where the scale and shape parameter are, respectively, the mean number of bursts per cell cycle and the mean number of <a href="/wiki/Protein_molecule" class="mw-redirect" title="Protein molecule">protein molecules</a> produced by a single mRNA during its lifetime.<sup id="cite_ref-Friedman_36-0" class="reference"><a href="#cite_note-Friedman-36">[36]</a></sup>
|
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</p><p>In <a href="/wiki/Genomics" title="Genomics">genomics</a>, the gamma distribution was applied in <a href="/wiki/Peak_calling" title="Peak calling">peak calling</a> step (i.e., in recognition of signal) in <a href="/wiki/ChIP-chip" class="mw-redirect" title="ChIP-chip">ChIP-chip</a><sup id="cite_ref-Reiss_37-0" class="reference"><a href="#cite_note-Reiss-37">[37]</a></sup> and <a href="/wiki/ChIP-seq" class="mw-redirect" title="ChIP-seq">ChIP-seq</a><sup id="cite_ref-Mendoza_38-0" class="reference"><a href="#cite_note-Mendoza-38">[38]</a></sup> data analysis.
|
|||
|
|
</p><p>In Bayesian statistics, the gamma distribution is widely used as a <a href="/wiki/Conjugate_prior" title="Conjugate prior">conjugate prior</a>. It is the conjugate prior for the <a href="/wiki/Precision_(statistics)" title="Precision (statistics)">precision</a> (i.e. inverse of the variance) of a <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>. It is also the conjugate prior for the <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a>.
|
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|
|
</p><p>In <a href="/wiki/Phylogenetics" title="Phylogenetics">phylogenetics</a>, the gamma distribution is the most commonly used approach to model among-sites rate variation<sup id="cite_ref-39" class="reference"><a href="#cite_note-39">[39]</a></sup> when <a href="/wiki/Computational_phylogenetics#Maximum_likelihood" title="Computational phylogenetics">maximum likelihood</a>, <a href="/wiki/Bayesian_inference_in_phylogeny" title="Bayesian inference in phylogeny">Bayesian</a>, or <a href="/wiki/Distance_matrices_in_phylogeny" title="Distance matrices in phylogeny">distance matrix methods</a> are used to estimate phylogenetic trees. Phylogenetic analyzes that use the gamma distribution to model rate variation estimate a single parameter from the data because they limit consideration to distributions where <i>α</i>=<i>β</i>. This parameterization means that the mean of this distribution is 1 and the variance is 1/<i>α</i>. Maximum likelihood and Bayesian methods typically use a discrete approximation to the continuous gamma distribution.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40">[40]</a></sup><sup id="cite_ref-41" class="reference"><a href="#cite_note-41">[41]</a></sup>
|
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</p>
|
|||
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<h2><span class="mw-headline" id="Random_variate_generation">Random variate generation</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=29" title="Edit section: Random variate generation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
|
|||
|
|
<p>Given the scaling property above, it is enough to generate gamma variables with <i>θ</i> = 1, as we can later convert to any value of <i>β</i> with a simple division.
|
|||
|
|
</p><p>Suppose we wish to generate random variables from Gamma(<i>n</i> + <i>δ</i>, 1), where n is a non-negative integer and 0 < <i>δ</i> < 1. Using the fact that a Gamma(1, 1) distribution is the same as an Exp(1) distribution, and noting the method of <a href="/wiki/Exponential_distribution#Random_variate_generation" title="Exponential distribution">generating exponential variables</a>, we conclude that if <i>U</i> is <a href="/wiki/Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">uniformly distributed</a> on (0, 1], then −ln(<i>U</i>) is distributed Gamma(1, 1) (i.e. <a href="/wiki/Inverse_transform_sampling" title="Inverse transform sampling">inverse transform sampling</a>). Now, using the "<i>α</i>-addition" property of gamma distribution, we expand this result:
|
|||
|
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</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\sum _{k=1}^{n}\ln U_{k}\sim \Gamma (n,1)}">
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
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|
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|
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|
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|
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|
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|
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|
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<annotation encoding="application/x-tex">{\displaystyle -\sum _{k=1}^{n}\ln U_{k}\sim \Gamma (n,1)}</annotation>
|
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|
|||
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88a83a9843f29cf470dc2831bda24053d4262bb9" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.892ex; height:6.843ex;" alt="{\displaystyle -\sum _{k=1}^{n}\ln U_{k}\sim \Gamma (n,1)}"></span></dd></dl>
|
|||
|
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<p>where <i>U</i><sub><i>k</i></sub> are all uniformly distributed on (0, 1] and <a href="/wiki/Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a>. All that is left now is to generate a variable distributed as Gamma(<i>δ</i>, 1) for 0 < <i>δ</i> < 1 and apply the "<i>α</i>-addition" property once more. This is the most difficult part.
|
|||
|
|
</p><p>Random generation of gamma variates is discussed in detail by Devroye,<sup id="cite_ref-devroye_42-0" class="reference"><a href="#cite_note-devroye-42">[42]</a></sup><sup class="reference nowrap"><span title="Page / location: 401–428">: 401–428 </span></sup> noting that none are uniformly fast for all shape parameters. For small values of the shape parameter, the algorithms are often not valid.<sup id="cite_ref-devroye_42-1" class="reference"><a href="#cite_note-devroye-42">[42]</a></sup><sup class="reference nowrap"><span title="Page / location: 406">: 406 </span></sup> For arbitrary values of the shape parameter, one can apply the Ahrens and Dieter<sup id="cite_ref-AD_43-0" class="reference"><a href="#cite_note-AD-43">[43]</a></sup> modified acceptance-rejection method Algorithm GD (shape <i>k</i> ≥ 1), or transformation method<sup id="cite_ref-44" class="reference"><a href="#cite_note-44">[44]</a></sup> when 0 < <i>k</i> < 1. Also see Cheng and Feast Algorithm GKM 3<sup id="cite_ref-45" class="reference"><a href="#cite_note-45">[45]</a></sup> or Marsaglia's squeeze method.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46">[46]</a></sup>
|
|||
|
|
</p><p>The following is a version of the Ahrens-Dieter <a href="/wiki/Rejection_sampling" title="Rejection sampling">acceptance–rejection method</a>:<sup id="cite_ref-AD_43-1" class="reference"><a href="#cite_note-AD-43">[43]</a></sup>
|
|||
|
|
</p>
|
|||
|
|
<ol><li>Generate <i>U</i>, <i>V</i> and <i>W</i> as <a href="/wiki/Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">iid</a> uniform (0, 1] variates.</li>
|
|||
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<li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\leq {\frac {e}{e+\delta }}}">
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|||
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|
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|
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|
|||
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|
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|
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|
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<annotation encoding="application/x-tex">{\displaystyle U\leq {\frac {e}{e+\delta }}}</annotation>
|
|||
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|
|||
|
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78687aeaefa95f26e5c2eedf29a5e9d16e354292" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:10.69ex; height:5.009ex;" alt="{\displaystyle U\leq {\frac {e}{e+\delta }}}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi =V^{1/\delta }}">
|
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|
|||
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|
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|
|||
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|
|||
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|
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|
|||
|
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
|
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|
|||
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|
|||
|
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|
|||
|
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|
|||
|
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|
|||
|
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|
|||
|
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<annotation encoding="application/x-tex">{\displaystyle \xi =V^{1/\delta }}</annotation>
|
|||
|
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|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc600e64975b15e012887f0072dd610e9ee928ab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.663ex; height:3.176ex;" alt="{\displaystyle \xi =V^{1/\delta }}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta =W\xi ^{\delta -1}}">
|
|||
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|
|||
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|
|||
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|
|||
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|
|||
|
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|
|||
|
|
<mi>W</mi>
|
|||
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|
|||
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|
|||
|
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|
|||
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|
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|
|||
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|
|||
|
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|
|||
|
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|
|||
|
|
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|
|||
|
|
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|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \eta =W\xi ^{\delta -1}}</annotation>
|
|||
|
|
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|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b9acf3fb974231242cb1622667fb3fef4dbd65b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.811ex; height:3.176ex;" alt="{\displaystyle \eta =W\xi ^{\delta -1}}"></span>. Otherwise, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi =1-\ln V}">
|
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
|
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|
|||
|
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<mi>ln</mi>
|
|||
|
|
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|
|||
|
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|
|||
|
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|
|||
|
|
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|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \xi =1-\ln V}</annotation>
|
|||
|
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|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01c3bb894921726acc9b91410489f8db898cddd5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.245ex; height:2.509ex;" alt="{\displaystyle \xi =1-\ln V}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta =We^{-\xi }}">
|
|||
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|
|||
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|
|||
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|
|||
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|
|||
|
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|
|||
|
|
<mi>W</mi>
|
|||
|
|
<msup>
|
|||
|
|
<mi>e</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>ξ<!-- ξ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \eta =We^{-\xi }}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6b92760858b460290e1acc45654f8713b8770ed" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.026ex; height:3.176ex;" alt="{\displaystyle \eta =We^{-\xi }}"></span>.</li>
|
|||
|
|
<li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta >\xi ^{\delta -1}e^{-\xi }}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>η<!-- η --></mi>
|
|||
|
|
<mo>></mo>
|
|||
|
|
<msup>
|
|||
|
|
<mi>ξ<!-- ξ --></mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>δ<!-- δ --></mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<msup>
|
|||
|
|
<mi>e</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>ξ<!-- ξ --></mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \eta >\xi ^{\delta -1}e^{-\xi }}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c999d5c04f901d0d448bed573cee6b42166dd46f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.699ex; height:3.176ex;" alt="{\displaystyle \eta >\xi ^{\delta -1}e^{-\xi }}"></span> then go to step 1.</li>
|
|||
|
|
<li><i>ξ</i> is distributed as Γ(<i>δ</i>, 1).</li></ol>
|
|||
|
|
<p>A summary of this is
|
|||
|
|
</p>
|
|||
|
|
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \left(\xi -\sum _{i=1}^{\lfloor k\rfloor }\ln(U_{i})\right)\sim \Gamma (k,\theta )}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mrow>
|
|||
|
|
<mo>(</mo>
|
|||
|
|
<mrow>
|
|||
|
|
<mi>ξ<!-- ξ --></mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<munderover>
|
|||
|
|
<mo>∑<!-- ∑ --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
</mrow>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mo fence="false" stretchy="false">⌊<!-- ⌊ --></mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo fence="false" stretchy="false">⌋<!-- ⌋ --></mo>
|
|||
|
|
</mrow>
|
|||
|
|
</munderover>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<msub>
|
|||
|
|
<mi>U</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mi>i</mi>
|
|||
|
|
</mrow>
|
|||
|
|
</msub>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>)</mo>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>∼<!-- ∼ --></mo>
|
|||
|
|
<mi mathvariant="normal">Γ<!-- Γ --></mi>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo>,</mo>
|
|||
|
|
<mi>θ<!-- θ --></mi>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \theta \left(\xi -\sum _{i=1}^{\lfloor k\rfloor }\ln(U_{i})\right)\sim \Gamma (k,\theta )}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/340448eee59052e9ed9066a5838215099dc9fa0f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.603ex; height:7.843ex;" alt="{\displaystyle \theta \left(\xi -\sum _{i=1}^{\lfloor k\rfloor }\ln(U_{i})\right)\sim \Gamma (k,\theta )}"></span></dd></dl>
|
|||
|
|
<p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \lfloor k\rfloor }">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mstyle displaystyle="false" scriptlevel="1">
|
|||
|
|
<mo fence="false" stretchy="false">⌊<!-- ⌊ --></mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
<mo fence="false" stretchy="false">⌋<!-- ⌋ --></mo>
|
|||
|
|
</mstyle>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \scriptstyle \lfloor k\rfloor }</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51f6297fc6736d117d930b473bc9d4325f8228ad" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.316ex; height:2.176ex;" alt="{\displaystyle \scriptstyle \lfloor k\rfloor }"></span> is the integer part of <i>k</i>, <i>ξ</i> is generated via the algorithm above with <i>δ</i> = {<i>k</i>} (the fractional part of <i>k</i>) and the <i>U</i><sub><i>k</i></sub> are all independent.
|
|||
|
|
</p><p>While the above approach is technically correct, Devroye notes that it is linear in the value of <i>k</i> and generally is not a good choice. Instead, he recommends using either rejection-based or table-based methods, depending on context.<sup id="cite_ref-devroye_42-2" class="reference"><a href="#cite_note-devroye-42">[42]</a></sup><sup class="reference nowrap"><span title="Page / location: 401–428">: 401–428 </span></sup>
|
|||
|
|
</p><p>For example, Marsaglia's simple transformation-rejection method relying on one normal variate <i>X</i> and one uniform variate <i>U</i>:<sup id="cite_ref-Marsaglia2000_21-1" class="reference"><a href="#cite_note-Marsaglia2000-21">[21]</a></sup>
|
|||
|
|
</p>
|
|||
|
|
<ol><li>Set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=a-{\frac {1}{3}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>d</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>a</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mn>3</mn>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle d=a-{\frac {1}{3}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b5f13d36077c9400af25d998bbe4bffa7772c5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.383ex; height:5.176ex;" alt="{\displaystyle d=a-{\frac {1}{3}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c={\frac {1}{\sqrt {9d}}}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>c</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<msqrt>
|
|||
|
|
<mn>9</mn>
|
|||
|
|
<mi>d</mi>
|
|||
|
|
</msqrt>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle c={\frac {1}{\sqrt {9d}}}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e74964f3d4c7c5a61d85bd18e36b9c6bedf0509d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:9.256ex; height:6.176ex;" alt="{\displaystyle c={\frac {1}{\sqrt {9d}}}}"></span>.</li>
|
|||
|
|
<li>Set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=(1+cX)^{3}}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>v</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mo stretchy="false">(</mo>
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mi>c</mi>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<msup>
|
|||
|
|
<mo stretchy="false">)</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>3</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle v=(1+cX)^{3}}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd133c8bb01c03e81f752a57d132ee59fcff2ea" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.079ex; height:3.176ex;" alt="{\displaystyle v=(1+cX)^{3}}"></span>.</li>
|
|||
|
|
<li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v>0}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>v</mi>
|
|||
|
|
<mo>></mo>
|
|||
|
|
<mn>0</mn>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle v>0}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c314fc908a83c555d34968d25e86c5ae0b76ef6f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.389ex; height:2.176ex;" alt="{\displaystyle v>0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln U<{\frac {X^{2}}{2}}+d-dv+d\ln v}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mi>U</mi>
|
|||
|
|
<mo><</mo>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mfrac>
|
|||
|
|
<msup>
|
|||
|
|
<mi>X</mi>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mrow>
|
|||
|
|
</msup>
|
|||
|
|
<mn>2</mn>
|
|||
|
|
</mfrac>
|
|||
|
|
</mrow>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mi>d</mi>
|
|||
|
|
<mo>−<!-- − --></mo>
|
|||
|
|
<mi>d</mi>
|
|||
|
|
<mi>v</mi>
|
|||
|
|
<mo>+</mo>
|
|||
|
|
<mi>d</mi>
|
|||
|
|
<mi>ln</mi>
|
|||
|
|
<mo>⁡<!-- --></mo>
|
|||
|
|
<mi>v</mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle \ln U<{\frac {X^{2}}{2}}+d-dv+d\ln v}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae59d8c5427259a4388fd99891925d4d8db16821" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.232ex; height:5.676ex;" alt="{\displaystyle \ln U<{\frac {X^{2}}{2}}+d-dv+d\ln v}"></span> return <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dv}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mi>d</mi>
|
|||
|
|
<mi>v</mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle dv}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e5f695cb4931398dd078f0335e6de6905abf748" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.343ex; height:2.176ex;" alt="dv"></span>, else go back to step 2.</li></ol>
|
|||
|
|
<p>With <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq a=\alpha =k}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
|
<mstyle displaystyle="true" scriptlevel="0">
|
|||
|
|
<mn>1</mn>
|
|||
|
|
<mo>≤<!-- ≤ --></mo>
|
|||
|
|
<mi>a</mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>α<!-- α --></mi>
|
|||
|
|
<mo>=</mo>
|
|||
|
|
<mi>k</mi>
|
|||
|
|
</mstyle>
|
|||
|
|
</mrow>
|
|||
|
|
<annotation encoding="application/x-tex">{\displaystyle 1\leq a=\alpha =k}</annotation>
|
|||
|
|
</semantics>
|
|||
|
|
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb8d12ad014d8646f91b7db9d53dd0a4bf1afdab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.387ex; height:2.343ex;" alt="1\leq a=\alpha =k"></span> generates a gamma distributed random number in time that is approximately constant with <i>k</i>. The acceptance rate does depend on <i>k</i>, with an acceptance rate of 0.95, 0.98, and 0.99 for k=1, 2, and 4. For <i>k</i> < 1, one can use <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{\alpha }=\gamma _{1+\alpha }U^{1/\alpha }}">
|
|||
|
|
<semantics>
|
|||
|
|
<mrow class="MJX-TeXAtom-ORD">
|
|||
|
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<annotation encoding="application/x-tex">{\displaystyle \gamma _{\alpha }=\gamma _{1+\alpha }U^{1/\alpha }}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a438686f43515309dc233e2bb3b2c8344357c7ef" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.945ex; height:3.343ex;" alt="{\displaystyle \gamma _{\alpha }=\gamma _{1+\alpha }U^{1/\alpha }}"></span> to boost <i>k</i> to be usable with this method.
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<h2><span class="mw-headline" id="References">References</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=30" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
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<style data-mw-deduplicate="TemplateStyles:r1011085734">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;">
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<ol class="references">
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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1133582631">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite id="CITEREFParkBera2009" class="citation journal cs1">Park, Sung Y.; Bera, Anil K. (2009). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160307144515/http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf">"Maximum entropy autoregressive conditional heteroskedasticity model"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of Econometrics</i>. <b>150</b> (2): 219–230. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.511.9750">10.1.1.511.9750</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jeconom.2008.12.014">10.1016/j.jeconom.2008.12.014</a>. Archived from <a rel="nofollow" class="external text" href="http://www.wise.xmu.edu.cn/Master/Download/..%5C..%5CUploadFiles%5Cpaper-masterdownload%5C2009519932327055475115776.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2016-03-07<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-06-02</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Econometrics&rft.atitle=Maximum+entropy+autoregressive+conditional+heteroskedasticity+model&rft.volume=150&rft.issue=2&rft.pages=219-230&rft.date=2009&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.511.9750%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1016%2Fj.jeconom.2008.12.014&rft.aulast=Park&rft.aufirst=Sung+Y.&rft.au=Bera%2C+Anil+K.&rft_id=http%3A%2F%2Fwww.wise.xmu.edu.cn%2FMaster%2FDownload%2F..%255C..%255CUploadFiles%255Cpaper-masterdownload%255C2009519932327055475115776.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span></span>
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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFHoggCraig1978" class="citation book cs1"><a href="/wiki/Robert_V._Hogg" title="Robert V. Hogg">Hogg, R. V.</a>; Craig, A. T. (1978). <i>Introduction to Mathematical Statistics</i> (4th ed.). New York: Macmillan. pp. Remark 3.3.1. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0023557109" title="Special:BookSources/0023557109"><bdi>0023557109</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Mathematical+Statistics&rft.place=New+York&rft.pages=Remark+3.3.1&rft.edition=4th&rft.pub=Macmillan&rft.date=1978&rft.isbn=0023557109&rft.aulast=Hogg&rft.aufirst=R.+V.&rft.au=Craig%2C+A.+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span></span>
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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFGopalanHofmanBlei2013" class="citation arxiv cs1">Gopalan, Prem; Hofman, Jake M.; <a href="/wiki/David_Blei" title="David Blei">Blei, David M.</a> (2013). "Scalable Recommendation with Poisson Factorization". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1311.1704">1311.1704</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.IR">cs.IR</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Scalable+Recommendation+with+Poisson+Factorization&rft.date=2013&rft_id=info%3Aarxiv%2F1311.1704&rft.aulast=Gopalan&rft.aufirst=Prem&rft.au=Hofman%2C+Jake+M.&rft.au=Blei%2C+David+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span></span>
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<li id="cite_note-Papoulis-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Papoulis_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Papoulis_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Papoulis, Pillai, <i>Probability, Random Variables, and Stochastic Processes</i>, Fourth Edition</span>
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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Jeesen Chen, <a href="/w/index.php?title=Herman_Rubin&action=edit&redlink=1" class="new" title="Herman Rubin (page does not exist)">Herman Rubin</a>, Bounds for the difference between median and mean of gamma and Poisson distributions, Statistics & Probability Letters, Volume 4, Issue 6, October 1986, Pages 281–283, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://www.worldcat.org/search?fq=x0:jrnl&q=n2:0167-7152">0167-7152</a>, <a rel="nofollow" class="external autonumber" href="https://dx.doi.org/10.1016/0167-7152(86)90044-1">[1]</a>.</span>
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<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Choi, K. P. <a rel="nofollow" class="external text" href="https://www.ams.org/journals/proc/1994-121-01/S0002-9939-1994-1195477-8/S0002-9939-1994-1195477-8.pdf">"On the Medians of the Gamma Distributions and an Equation of Ramanujan"</a>, Proceedings of the American Mathematical Society, Vol. 121, No. 1 (May, 1994), pp. 245–251.</span>
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<li id="cite_note-berg-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-berg_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-berg_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFBerg,_ChristianPedersen,_Henrik_L.2006" class="citation journal cs1">Berg, Christian & Pedersen, Henrik L. (March 2006). <a rel="nofollow" class="external text" href="https://www.intlpress.com/site/pub/files/_fulltext/journals/maa/2006/0013/0001/MAA-2006-0013-0001-a004.pdf">"The Chen–Rubin conjecture in a continuous setting"</a> <span class="cs1-format">(PDF)</span>. <i>Methods and Applications of Analysis</i>. <b>13</b> (1): 63–88. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4310%2FMAA.2006.v13.n1.a4">10.4310/MAA.2006.v13.n1.a4</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6704865">6704865</a><span class="reference-accessdate">. Retrieved <span class="nowrap">1 April</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Methods+and+Applications+of+Analysis&rft.atitle=The+Chen%E2%80%93Rubin+conjecture+in+a+continuous+setting&rft.volume=13&rft.issue=1&rft.pages=63-88&rft.date=2006-03&rft_id=info%3Adoi%2F10.4310%2FMAA.2006.v13.n1.a4&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A6704865%23id-name%3DS2CID&rft.au=Berg%2C+Christian&rft.au=Pedersen%2C+Henrik+L.&rft_id=https%3A%2F%2Fwww.intlpress.com%2Fsite%2Fpub%2Ffiles%2F_fulltext%2Fjournals%2Fmaa%2F2006%2F0013%2F0001%2FMAA-2006-0013-0001-a004.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span></span>
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<li id="cite_note-convex-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-convex_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-convex_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Berg, Christian and Pedersen, Henrik L. <a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0609442">"Convexity of the median in the gamma distribution"</a>.</span>
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<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFGaunt,_Robert_E.,_and_Milan_Merkle2021" class="citation journal cs1">Gaunt, Robert E., and Milan Merkle (2021). "On bounds for the mode and median of the generalized hyperbolic and related distributions". <i>Journal of Mathematical Analysis and Applications</i>. <b>493</b> (1): 124508. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2002.01884">2002.01884</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jmaa.2020.124508">10.1016/j.jmaa.2020.124508</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:221103640">221103640</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Mathematical+Analysis+and+Applications&rft.atitle=On+bounds+for+the+mode+and+median+of+the+generalized+hyperbolic+and+related+distributions&rft.volume=493&rft.issue=1&rft.pages=124508&rft.date=2021&rft_id=info%3Aarxiv%2F2002.01884&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A221103640%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.jmaa.2020.124508&rft.au=Gaunt%2C+Robert+E.%2C+and+Milan+Merkle&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span>
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<li id="cite_note-lyon-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-lyon_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-lyon_10-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-lyon_10-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFLyon2021" class="citation journal cs1">Lyon, Richard F. (13 May 2021). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8118309">"On closed-form tight bounds and approximations for the median of a gamma distribution"</a>. <i><a href="/wiki/PLOS_One" title="PLOS One">PLOS One</a></i>. <b>16</b> (5): e0251626. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2011.04060">2011.04060</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2021PLoSO..1651626L">2021PLoSO..1651626L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1371%2Fjournal.pone.0251626">10.1371/journal.pone.0251626</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8118309">8118309</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/33984053">33984053</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=PLOS+One&rft.atitle=On+closed-form+tight+bounds+and+approximations+for+the+median+of+a+gamma+distribution&rft.volume=16&rft.issue=5&rft.pages=e0251626&rft.date=2021-05-13&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC8118309%23id-name%3DPMC&rft_id=info%3Abibcode%2F2021PLoSO..1651626L&rft_id=info%3Aarxiv%2F2011.04060&rft_id=info%3Apmid%2F33984053&rft_id=info%3Adoi%2F10.1371%2Fjournal.pone.0251626&rft.aulast=Lyon&rft.aufirst=Richard+F.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC8118309&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span></span>
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<li id="cite_note-lyon2023-11"><span class="mw-cite-backlink">^ <a href="#cite_ref-lyon2023_11-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-lyon2023_11-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFLyon2021" class="citation journal cs1">Lyon, Richard F. (13 May 2021). <a rel="nofollow" class="external text" href="https://doi.org/10.1371%2Fjournal.pone.0288601">"Tight bounds for the median of a gamma distribution"</a>. <i><a href="/wiki/PLOS_One" title="PLOS One">PLOS One</a></i>. <b>18</b> (9): e0288601. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1371%2Fjournal.pone.0288601">10.1371/journal.pone.0288601</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=PLOS+One&rft.atitle=Tight+bounds+for+the+median+of+a+gamma+distribution&rft.volume=18&rft.issue=9&rft.pages=e0288601&rft.date=2021-05-13&rft_id=info%3Adoi%2F10.1371%2Fjournal.pone.0288601&rft.aulast=Lyon&rft.aufirst=Richard+F.&rft_id=https%3A%2F%2Fdoi.org%2F10.1371%252Fjournal.pone.0288601&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span></span>
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<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFMathai1982" class="citation journal cs1">Mathai, A. M. (1982). "Storage capacity of a dam with gamma type inputs". <i>Annals of the Institute of Statistical Mathematics</i>. <b>34</b> (3): 591–597. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02481056">10.1007/BF02481056</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://www.worldcat.org/issn/0020-3157">0020-3157</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122537756">122537756</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+the+Institute+of+Statistical+Mathematics&rft.atitle=Storage+capacity+of+a+dam+with+gamma+type+inputs&rft.volume=34&rft.issue=3&rft.pages=591-597&rft.date=1982&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122537756%23id-name%3DS2CID&rft.issn=0020-3157&rft_id=info%3Adoi%2F10.1007%2FBF02481056&rft.aulast=Mathai&rft.aufirst=A.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span></span>
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<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFMoschopoulos1985" class="citation journal cs1">Moschopoulos, P. G. (1985). "The distribution of the sum of independent gamma random variables". <i>Annals of the Institute of Statistical Mathematics</i>. <b>37</b> (3): 541–544. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02481123">10.1007/BF02481123</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120066454">120066454</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+the+Institute+of+Statistical+Mathematics&rft.atitle=The+distribution+of+the+sum+of+independent+gamma+random+variables&rft.volume=37&rft.issue=3&rft.pages=541-544&rft.date=1985&rft_id=info%3Adoi%2F10.1007%2FBF02481123&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120066454%23id-name%3DS2CID&rft.aulast=Moschopoulos&rft.aufirst=P.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span></span>
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<li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">W.D. Penny, [www.fil.ion.ucl.ac.uk/~wpenny/publications/densities.ps KL-Divergences of Normal, Gamma, Dirichlet, and Wishart densities]<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources#What_information_to_include" title="Wikipedia:Citing sources"><span title="A complete citation is needed. (November 2012)">full citation needed</span></a></i>]</sup></span>
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<li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://reference.wolfram.com/language/ref/ExpGammaDistribution.html">"ExpGammaDistribution—Wolfram Language Documentation"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=ExpGammaDistribution%E2%80%94Wolfram+Language+Documentation&rft_id=https%3A%2F%2Freference.wolfram.com%2Flanguage%2Fref%2FExpGammaDistribution.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span></span>
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<li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.loggamma.html#scipy.stats.loggamma">"scipy.stats.loggamma — SciPy v1.8.0 Manual"</a>. <i>docs.scipy.org</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=docs.scipy.org&rft.atitle=scipy.stats.loggamma+%E2%80%94+SciPy+v1.8.0+Manual&rft_id=https%3A%2F%2Fdocs.scipy.org%2Fdoc%2Fscipy%2Freference%2Fgenerated%2Fscipy.stats.loggamma.html%23scipy.stats.loggamma&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span></span>
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<li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFSunKongPal2021" class="citation journal cs1">Sun, Jingchao; Kong, Maiying; Pal, Subhadip (22 June 2021). <a rel="nofollow" class="external text" href="https://www.tandfonline.com/doi/full/10.1080/03610926.2021.1934700?scroll=top&needAccess=true">"The Modified-Half-Normal distribution: Properties and an efficient sampling scheme"</a>. <i>Communications in Statistics - Theory and Methods</i>. <b>52</b> (5): 1591–1613. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F03610926.2021.1934700">10.1080/03610926.2021.1934700</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://www.worldcat.org/issn/0361-0926">0361-0926</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:237919587">237919587</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+in+Statistics+-+Theory+and+Methods&rft.atitle=The+Modified-Half-Normal+distribution%3A+Properties+and+an+efficient+sampling+scheme&rft.volume=52&rft.issue=5&rft.pages=1591-1613&rft.date=2021-06-22&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A237919587%23id-name%3DS2CID&rft.issn=0361-0926&rft_id=info%3Adoi%2F10.1080%2F03610926.2021.1934700&rft.aulast=Sun&rft.aufirst=Jingchao&rft.au=Kong%2C+Maiying&rft.au=Pal%2C+Subhadip&rft_id=https%3A%2F%2Fwww.tandfonline.com%2Fdoi%2Ffull%2F10.1080%2F03610926.2021.1934700%3Fscroll%3Dtop%26needAccess%3Dtrue&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span></span>
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<li id="cite_note-Dubey-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-Dubey_18-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFDubey1970" class="citation journal cs1">Dubey, Satya D. (December 1970). "Compound gamma, beta and F distributions". <i>Metrika</i>. <b>16</b>: 27–31. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02613934">10.1007/BF02613934</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123366328">123366328</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Metrika&rft.atitle=Compound+gamma%2C+beta+and+F+distributions&rft.volume=16&rft.pages=27-31&rft.date=1970-12&rft_id=info%3Adoi%2F10.1007%2FBF02613934&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123366328%23id-name%3DS2CID&rft.aulast=Dubey&rft.aufirst=Satya+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span></span>
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<li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFMinka2002" class="citation journal cs1">Minka, Thomas P. (2002). <a rel="nofollow" class="external text" href="https://tminka.github.io/papers/minka-gamma.pdf">"Estimating a Gamma distribution"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Estimating+a+Gamma+distribution&rft.date=2002&rft.aulast=Minka&rft.aufirst=Thomas+P.&rft_id=https%3A%2F%2Ftminka.github.io%2Fpapers%2Fminka-gamma.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: </span><span class="cs1-visible-error citation-comment">Cite journal requires <code class="cs1-code">|journal=</code> (<a href="/wiki/Help:CS1_errors#missing_periodical" title="Help:CS1 errors">help</a>)</span></span>
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<li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFChoiWette1969" class="citation journal cs1">Choi, S. C.; Wette, R. (1969). "Maximum Likelihood Estimation of the Parameters of the Gamma Distribution and Their Bias". <i>Technometrics</i>. <b>11</b> (4): 683–690. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00401706.1969.10490731">10.1080/00401706.1969.10490731</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Technometrics&rft.atitle=Maximum+Likelihood+Estimation+of+the+Parameters+of+the+Gamma+Distribution+and+Their+Bias&rft.volume=11&rft.issue=4&rft.pages=683-690&rft.date=1969&rft_id=info%3Adoi%2F10.1080%2F00401706.1969.10490731&rft.aulast=Choi&rft.aufirst=S.+C.&rft.au=Wette%2C+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span></span>
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<li id="cite_note-Marsaglia2000-21"><span class="mw-cite-backlink">^ <a href="#cite_ref-Marsaglia2000_21-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Marsaglia2000_21-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFMarsagliaTsang2000" class="citation journal cs1">Marsaglia, G.; Tsang, W. W. (2000). "A simple method for generating gamma variables". <i>ACM Transactions on Mathematical Software</i>. <b>26</b> (3): 363–372. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F358407.358414">10.1145/358407.358414</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2634158">2634158</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ACM+Transactions+on+Mathematical+Software&rft.atitle=A+simple+method+for+generating+gamma+variables&rft.volume=26&rft.issue=3&rft.pages=363-372&rft.date=2000&rft_id=info%3Adoi%2F10.1145%2F358407.358414&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A2634158%23id-name%3DS2CID&rft.aulast=Marsaglia&rft.aufirst=G.&rft.au=Tsang%2C+W.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span></span>
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<li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFYeChen2017" class="citation journal cs1">Ye, Zhi-Sheng; Chen, Nan (2017). <a rel="nofollow" class="external text" href="https://amstat.tandfonline.com/doi/abs/10.1080/00031305.2016.1209129">"Closed-Form Estimators for the Gamma Distribution Derived from Likelihood Equations"</a>. <i>The American Statistician</i>. <b>71</b> (2): 177–181. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00031305.2016.1209129">10.1080/00031305.2016.1209129</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:124682698">124682698</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Statistician&rft.atitle=Closed-Form+Estimators+for+the+Gamma+Distribution+Derived+from+Likelihood+Equations&rft.volume=71&rft.issue=2&rft.pages=177-181&rft.date=2017&rft_id=info%3Adoi%2F10.1080%2F00031305.2016.1209129&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A124682698%23id-name%3DS2CID&rft.aulast=Ye&rft.aufirst=Zhi-Sheng&rft.au=Chen%2C+Nan&rft_id=https%3A%2F%2Famstat.tandfonline.com%2Fdoi%2Fabs%2F10.1080%2F00031305.2016.1209129&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span></span>
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<li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFLouzadaRamosRamos2019" class="citation journal cs1">Louzada, Francisco; Ramos, Pedro L.; Ramos, Eduardo (2019). <a rel="nofollow" class="external text" href="https://www.tandfonline.com/doi/abs/10.1080/00031305.2018.1513376">"A Note on Bias of Closed-Form Estimators for the Gamma Distribution Derived from Likelihood Equations"</a>. <i>The American Statistician</i>. <b>73</b> (2): 195–199. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00031305.2018.1513376">10.1080/00031305.2018.1513376</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:126086375">126086375</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Statistician&rft.atitle=A+Note+on+Bias+of+Closed-Form+Estimators+for+the+Gamma+Distribution+Derived+from+Likelihood+Equations&rft.volume=73&rft.issue=2&rft.pages=195-199&rft.date=2019&rft_id=info%3Adoi%2F10.1080%2F00031305.2018.1513376&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A126086375%23id-name%3DS2CID&rft.aulast=Louzada&rft.aufirst=Francisco&rft.au=Ramos%2C+Pedro+L.&rft.au=Ramos%2C+Eduardo&rft_id=https%3A%2F%2Fwww.tandfonline.com%2Fdoi%2Fabs%2F10.1080%2F00031305.2018.1513376&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span></span>
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<li id="cite_note-fink-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-fink_24-0">^</a></b></span> <span class="reference-text">Fink, D. 1995 <a rel="nofollow" class="external text" href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.157.5540&rep=rep1&type=pdf">A Compendium of Conjugate Priors</a>. In progress report: Extension and enhancement of methods for setting data quality objectives. (DOE contract 95‑831).</span>
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<li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFJessica.2001" class="citation book cs1">Jessica., Scheiner, Samuel M., 1956- Gurevitch (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AgsTDAAAQBAJ&dq=gamma+distribution+failure+waiting+time&pg=PA235">"13. Failure-time analysis"</a>. <a rel="nofollow" class="external text" href="http://worldcat.org/oclc/43694448"><i>Design and analysis of ecological experiments</i></a>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-513187-8" title="Special:BookSources/0-19-513187-8"><bdi>0-19-513187-8</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/43694448">43694448</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=13.+Failure-time+analysis&rft.btitle=Design+and+analysis+of+ecological+experiments&rft.pub=Oxford+University+Press&rft.date=2001&rft_id=info%3Aoclcnum%2F43694448&rft.isbn=0-19-513187-8&rft.aulast=Jessica.&rft.aufirst=Scheiner%2C+Samuel+M.%2C+1956-+Gurevitch&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DAgsTDAAAQBAJ%26dq%3Dgamma%2Bdistribution%2Bfailure%2Bwaiting%2Btime%26pg%3DPA235&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>) CS1 maint: numeric names: authors list (<a href="/wiki/Category:CS1_maint:_numeric_names:_authors_list" title="Category:CS1 maint: numeric names: authors list">link</a>)</span></span>
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<li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFPoonDavisChao2005" class="citation journal cs1">Poon, Art; Davis, Bradley H; Chao, Lin (2005-07-01). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1534/genetics.104.037259">"The Coupon Collector and the Suppressor Mutation"</a>. <i>Genetics</i>. <b>170</b> (3): 1323–1332. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1534%2Fgenetics.104.037259">10.1534/genetics.104.037259</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://www.worldcat.org/issn/1943-2631">1943-2631</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1451182">1451182</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/15879511">15879511</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Genetics&rft.atitle=The+Coupon+Collector+and+the+Suppressor+Mutation&rft.volume=170&rft.issue=3&rft.pages=1323-1332&rft.date=2005-07-01&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1451182%23id-name%3DPMC&rft.issn=1943-2631&rft_id=info%3Apmid%2F15879511&rft_id=info%3Adoi%2F10.1534%2Fgenetics.104.037259&rft.aulast=Poon&rft.aufirst=Art&rft.au=Davis%2C+Bradley+H&rft.au=Chao%2C+Lin&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1534%2Fgenetics.104.037259&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span></span>
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<li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFBelikov2017" class="citation journal cs1">Belikov, Aleksey V. (22 September 2017). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5610194">"The number of key carcinogenic events can be predicted from cancer incidence"</a>. <i>Scientific Reports</i>. <b>7</b> (1): 12170. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017NatSR...712170B">2017NatSR...712170B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fs41598-017-12448-7">10.1038/s41598-017-12448-7</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5610194">5610194</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/28939880">28939880</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+Reports&rft.atitle=The+number+of+key+carcinogenic+events+can+be+predicted+from+cancer+incidence&rft.volume=7&rft.issue=1&rft.pages=12170&rft.date=2017-09-22&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5610194%23id-name%3DPMC&rft_id=info%3Apmid%2F28939880&rft_id=info%3Adoi%2F10.1038%2Fs41598-017-12448-7&rft_id=info%3Abibcode%2F2017NatSR...712170B&rft.aulast=Belikov&rft.aufirst=Aleksey+V.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5610194&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span></span>
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<li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFYang1994" class="citation journal cs1">Yang, Ziheng (September 1994). <a rel="nofollow" class="external text" href="http://link.springer.com/10.1007/BF00160154">"Maximum likelihood phylogenetic estimation from DNA sequences with variable rates over sites: Approximate methods"</a>. <i>Journal of Molecular Evolution</i>. <b>39</b> (3): 306–314. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1994JMolE..39..306Y">1994JMolE..39..306Y</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00160154">10.1007/BF00160154</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://www.worldcat.org/issn/0022-2844">0022-2844</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/7932792">7932792</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17911050">17911050</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Molecular+Evolution&rft.atitle=Maximum+likelihood+phylogenetic+estimation+from+DNA+sequences+with+variable+rates+over+sites%3A+Approximate+methods&rft.volume=39&rft.issue=3&rft.pages=306-314&rft.date=1994-09&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17911050%23id-name%3DS2CID&rft_id=info%3Abibcode%2F1994JMolE..39..306Y&rft.issn=0022-2844&rft_id=info%3Adoi%2F10.1007%2FBF00160154&rft_id=info%3Apmid%2F7932792&rft.aulast=Yang&rft.aufirst=Ziheng&rft_id=http%3A%2F%2Flink.springer.com%2F10.1007%2FBF00160154&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span></span>
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<li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFFelsenstein2001" class="citation journal cs1">Felsenstein, Joseph (2001-10-01). <a rel="nofollow" class="external text" href="http://link.springer.com/10.1007/s002390010234">"Taking Variation of Evolutionary Rates Between Sites into Account in Inferring Phylogenies"</a>. <i>Journal of Molecular Evolution</i>. <b>53</b> (4–5): 447–455. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001JMolE..53..447F">2001JMolE..53..447F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs002390010234">10.1007/s002390010234</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://www.worldcat.org/issn/0022-2844">0022-2844</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11675604">11675604</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:9791493">9791493</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Molecular+Evolution&rft.atitle=Taking+Variation+of+Evolutionary+Rates+Between+Sites+into+Account+in+Inferring+Phylogenies&rft.volume=53&rft.issue=4%E2%80%935&rft.pages=447-455&rft.date=2001-10-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A9791493%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2001JMolE..53..447F&rft.issn=0022-2844&rft_id=info%3Adoi%2F10.1007%2Fs002390010234&rft_id=info%3Apmid%2F11675604&rft.aulast=Felsenstein&rft.aufirst=Joseph&rft_id=http%3A%2F%2Flink.springer.com%2F10.1007%2Fs002390010234&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span></span>
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<li id="cite_note-devroye-42"><span class="mw-cite-backlink">^ <a href="#cite_ref-devroye_42-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-devroye_42-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-devroye_42-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFDevroye1986" class="citation book cs1">Devroye, Luc (1986). <a rel="nofollow" class="external text" href="http://luc.devroye.org/rnbookindex.html"><i>Non-Uniform Random Variate Generation</i></a>. New York: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-96305-1" title="Special:BookSources/978-0-387-96305-1"><bdi>978-0-387-96305-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Non-Uniform+Random+Variate+Generation&rft.place=New+York&rft.pub=Springer-Verlag&rft.date=1986&rft.isbn=978-0-387-96305-1&rft.aulast=Devroye&rft.aufirst=Luc&rft_id=http%3A%2F%2Fluc.devroye.org%2Frnbookindex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span> See Chapter 9, Section 3.</span>
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<li id="cite_note-AD-43"><span class="mw-cite-backlink">^ <a href="#cite_ref-AD_43-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-AD_43-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFAhrensDieter1982" class="citation journal cs1">Ahrens, J. H.; Dieter, U (January 1982). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F358315.358390">"Generating gamma variates by a modified rejection technique"</a>. <i>Communications of the ACM</i>. <b>25</b> (1): 47–54. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F358315.358390">10.1145/358315.358390</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15128188">15128188</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+of+the+ACM&rft.atitle=Generating+gamma+variates+by+a+modified+rejection+technique&rft.volume=25&rft.issue=1&rft.pages=47-54&rft.date=1982-01&rft_id=info%3Adoi%2F10.1145%2F358315.358390&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15128188%23id-name%3DS2CID&rft.aulast=Ahrens&rft.aufirst=J.+H.&rft.au=Dieter%2C+U&rft_id=https%3A%2F%2Fdoi.org%2F10.1145%252F358315.358390&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span>. See Algorithm GD, p. 53.</span>
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<li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFAhrensDieter1974" class="citation journal cs1">Ahrens, J. H.; Dieter, U. (1974). "Computer methods for sampling from gamma, beta, Poisson and binomial distributions". <i>Computing</i>. <b>12</b> (3): 223–246. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.93.3828">10.1.1.93.3828</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02293108">10.1007/BF02293108</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:37484126">37484126</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computing&rft.atitle=Computer+methods+for+sampling+from+gamma%2C+beta%2C+Poisson+and+binomial+distributions&rft.volume=12&rft.issue=3&rft.pages=223-246&rft.date=1974&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.93.3828%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A37484126%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF02293108&rft.aulast=Ahrens&rft.aufirst=J.+H.&rft.au=Dieter%2C+U.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span></span>
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<li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFChengFeast1979" class="citation journal cs1">Cheng, R. C. H.; Feast, G. M. (1979). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2347200">"Some Simple Gamma Variate Generators"</a>. <i>Journal of the Royal Statistical Society. Series C (Applied Statistics)</i>. <b>28</b> (3): 290–295. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2347200">10.2307/2347200</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2347200">2347200</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Royal+Statistical+Society.+Series+C+%28Applied+Statistics%29&rft.atitle=Some+Simple+Gamma+Variate+Generators&rft.volume=28&rft.issue=3&rft.pages=290-295&rft.date=1979&rft_id=info%3Adoi%2F10.2307%2F2347200&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2347200%23id-name%3DJSTOR&rft.aulast=Cheng&rft.aufirst=R.+C.+H.&rft.au=Feast%2C+G.+M.&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2347200&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span></span>
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<li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text">Marsaglia, G. The squeeze method for generating gamma variates. Comput, Math. Appl. 3 (1977), 321–325.</span>
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<h2><span class="mw-headline" id="External_links">External links</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gamma_distribution&action=edit&section=31" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
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<div class="side-box-text plainlist">The Wikibook <i><a href="https://en.wikibooks.org/wiki/Statistics" class="extiw" title="wikibooks:Statistics">Statistics</a></i> has a page on the topic of: <i><b><a href="https://en.wikibooks.org/wiki/Statistics/Distributions/Gamma" class="extiw" title="wikibooks:Statistics/Distributions/Gamma">Gamma distribution</a></b></i></div></div>
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<ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Gamma-distribution">"Gamma-distribution"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Gamma-distribution&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DGamma-distribution&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span></li>
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<li><span class="citation mathworld" id="Reference-Mathworld-Gamma_distribution"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/GammaDistribution.html">"Gamma distribution"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Gamma+distribution&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FGammaDistribution.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></span></span></li>
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<li>ModelAssist (2017) <a rel="nofollow" class="external text" href="http://www.epixanalytics.com/modelassist/AtRisk/Model_Assist.htm#Distributions/Continuous_distributions/Gamma.htm">Uses of the gamma distribution in risk modeling, including applied examples in Excel</a>.</li>
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<li><a rel="nofollow" class="external text" href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda366b.htm">Engineering Statistics Handbook</a></li></ul>
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<div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1061467846">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-pa
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<ul><li><a href="/wiki/Benford%27s_law" title="Benford's law">Benford</a></li>
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<li><a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli</a></li>
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<li><a href="/wiki/Beta-binomial_distribution" title="Beta-binomial distribution">beta-binomial</a></li>
|
|||
|
|
<li><a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial</a></li>
|
|||
|
|
<li><a href="/wiki/Categorical_distribution" title="Categorical distribution">categorical</a></li>
|
|||
|
|
<li><a href="/wiki/Hypergeometric_distribution" title="Hypergeometric distribution">hypergeometric</a>
|
|||
|
|
<ul><li><a href="/wiki/Negative_hypergeometric_distribution" title="Negative hypergeometric distribution">negative</a></li></ul></li>
|
|||
|
|
<li><a href="/wiki/Poisson_binomial_distribution" title="Poisson binomial distribution">Poisson binomial</a></li>
|
|||
|
|
<li><a href="/wiki/Rademacher_distribution" title="Rademacher distribution">Rademacher</a></li>
|
|||
|
|
<li><a href="/wiki/Soliton_distribution" title="Soliton distribution">soliton</a></li>
|
|||
|
|
<li><a href="/wiki/Discrete_uniform_distribution" title="Discrete uniform distribution">discrete uniform</a></li>
|
|||
|
|
<li><a href="/wiki/Zipf%27s_law" title="Zipf's law">Zipf</a></li>
|
|||
|
|
<li><a href="/wiki/Zipf%E2%80%93Mandelbrot_law" title="Zipf–Mandelbrot law">Zipf–Mandelbrot</a></li></ul>
|
|||
|
|
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">with infinite <br />support</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
|
|||
|
|
<ul><li><a href="/wiki/Beta_negative_binomial_distribution" title="Beta negative binomial distribution">beta negative binomial</a></li>
|
|||
|
|
<li><a href="/wiki/Borel_distribution" title="Borel distribution">Borel</a></li>
|
|||
|
|
<li><a href="/wiki/Conway%E2%80%93Maxwell%E2%80%93Poisson_distribution" title="Conway–Maxwell–Poisson distribution">Conway–Maxwell–Poisson</a></li>
|
|||
|
|
<li><a href="/wiki/Discrete_phase-type_distribution" title="Discrete phase-type distribution">discrete phase-type</a></li>
|
|||
|
|
<li><a href="/wiki/Delaporte_distribution" title="Delaporte distribution">Delaporte</a></li>
|
|||
|
|
<li><a href="/wiki/Extended_negative_binomial_distribution" title="Extended negative binomial distribution">extended negative binomial</a></li>
|
|||
|
|
<li><a href="/wiki/Flory%E2%80%93Schulz_distribution" title="Flory–Schulz distribution">Flory–Schulz</a></li>
|
|||
|
|
<li><a href="/wiki/Gauss%E2%80%93Kuzmin_distribution" title="Gauss–Kuzmin distribution">Gauss–Kuzmin</a></li>
|
|||
|
|
<li><a href="/wiki/Geometric_distribution" title="Geometric distribution">geometric</a></li>
|
|||
|
|
<li><a href="/wiki/Logarithmic_distribution" title="Logarithmic distribution">logarithmic</a></li>
|
|||
|
|
<li><a href="/wiki/Mixed_Poisson_distribution" title="Mixed Poisson distribution">mixed Poisson</a></li>
|
|||
|
|
<li><a href="/wiki/Negative_binomial_distribution" title="Negative binomial distribution">negative binomial</a></li>
|
|||
|
|
<li><a href="/wiki/(a,b,0)_class_of_distributions" title="(a,b,0) class of distributions">Panjer</a></li>
|
|||
|
|
<li><a href="/wiki/Parabolic_fractal_distribution" title="Parabolic fractal distribution">parabolic fractal</a></li>
|
|||
|
|
<li><a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson</a></li>
|
|||
|
|
<li><a href="/wiki/Skellam_distribution" title="Skellam distribution">Skellam</a></li>
|
|||
|
|
<li><a href="/wiki/Yule%E2%80%93Simon_distribution" title="Yule–Simon distribution">Yule–Simon</a></li>
|
|||
|
|
<li><a href="/wiki/Zeta_distribution" title="Zeta distribution">zeta</a></li></ul>
|
|||
|
|
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Continuous <br />univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">supported on a <br />bounded interval</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
|
|||
|
|
<ul><li><a href="/wiki/Arcsine_distribution" title="Arcsine distribution">arcsine</a></li>
|
|||
|
|
<li><a href="/wiki/ARGUS_distribution" title="ARGUS distribution">ARGUS</a></li>
|
|||
|
|
<li><a href="/wiki/Balding%E2%80%93Nichols_model" title="Balding–Nichols model">Balding–Nichols</a></li>
|
|||
|
|
<li><a href="/wiki/Bates_distribution" title="Bates distribution">Bates</a></li>
|
|||
|
|
<li><a href="/wiki/Beta_distribution" title="Beta distribution">beta</a></li>
|
|||
|
|
<li><a href="/wiki/Beta_rectangular_distribution" title="Beta rectangular distribution">beta rectangular</a></li>
|
|||
|
|
<li><a href="/wiki/Continuous_Bernoulli_distribution" title="Continuous Bernoulli distribution">continuous Bernoulli</a></li>
|
|||
|
|
<li><a href="/wiki/Irwin%E2%80%93Hall_distribution" title="Irwin–Hall distribution">Irwin–Hall</a></li>
|
|||
|
|
<li><a href="/wiki/Kumaraswamy_distribution" title="Kumaraswamy distribution">Kumaraswamy</a></li>
|
|||
|
|
<li><a href="/wiki/Logit-normal_distribution" title="Logit-normal distribution">logit-normal</a></li>
|
|||
|
|
<li><a href="/wiki/Noncentral_beta_distribution" title="Noncentral beta distribution">noncentral beta</a></li>
|
|||
|
|
<li><a href="/wiki/PERT_distribution" title="PERT distribution">PERT</a></li>
|
|||
|
|
<li><a href="/wiki/Raised_cosine_distribution" title="Raised cosine distribution">raised cosine</a></li>
|
|||
|
|
<li><a href="/wiki/Reciprocal_distribution" title="Reciprocal distribution">reciprocal</a></li>
|
|||
|
|
<li><a href="/wiki/Triangular_distribution" title="Triangular distribution">triangular</a></li>
|
|||
|
|
<li><a href="/wiki/U-quadratic_distribution" title="U-quadratic distribution">U-quadratic</a></li>
|
|||
|
|
<li><a href="/wiki/Continuous_uniform_distribution" title="Continuous uniform distribution">uniform</a></li>
|
|||
|
|
<li><a href="/wiki/Wigner_semicircle_distribution" title="Wigner semicircle distribution">Wigner semicircle</a></li></ul>
|
|||
|
|
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">supported on a <br />semi-infinite <br />interval</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
|
|||
|
|
<ul><li><a href="/wiki/Benini_distribution" title="Benini distribution">Benini</a></li>
|
|||
|
|
<li><a href="/wiki/Benktander_type_I_distribution" title="Benktander type I distribution">Benktander 1st kind</a></li>
|
|||
|
|
<li><a href="/wiki/Benktander_type_II_distribution" title="Benktander type II distribution">Benktander 2nd kind</a></li>
|
|||
|
|
<li><a href="/wiki/Beta_prime_distribution" title="Beta prime distribution">beta prime</a></li>
|
|||
|
|
<li><a href="/wiki/Burr_distribution" title="Burr distribution">Burr</a></li>
|
|||
|
|
<li><a href="/wiki/Chi_distribution" title="Chi distribution">chi</a></li>
|
|||
|
|
<li><a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">chi-squared</a>
|
|||
|
|
<ul><li><a href="/wiki/Noncentral_chi-squared_distribution" title="Noncentral chi-squared distribution">noncentral</a></li>
|
|||
|
|
<li><a href="/wiki/Inverse-chi-squared_distribution" title="Inverse-chi-squared distribution">inverse</a>
|
|||
|
|
<ul><li><a href="/wiki/Scaled_inverse_chi-squared_distribution" title="Scaled inverse chi-squared distribution">scaled</a></li></ul></li></ul></li>
|
|||
|
|
<li><a href="/wiki/Dagum_distribution" title="Dagum distribution">Dagum</a></li>
|
|||
|
|
<li><a href="/wiki/Davis_distribution" title="Davis distribution">Davis</a></li>
|
|||
|
|
<li><a href="/wiki/Erlang_distribution" title="Erlang distribution">Erlang</a>
|
|||
|
|
<ul><li><a href="/wiki/Hyper-Erlang_distribution" title="Hyper-Erlang distribution">hyper</a></li></ul></li>
|
|||
|
|
<li><a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential</a>
|
|||
|
|
<ul><li><a href="/wiki/Hyperexponential_distribution" title="Hyperexponential distribution">hyperexponential</a></li>
|
|||
|
|
<li><a href="/wiki/Hypoexponential_distribution" title="Hypoexponential distribution">hypoexponential</a></li>
|
|||
|
|
<li><a href="/wiki/Exponential-logarithmic_distribution" title="Exponential-logarithmic distribution">logarithmic</a></li></ul></li>
|
|||
|
|
<li><a href="/wiki/F-distribution" title="F-distribution"><i>F</i></a>
|
|||
|
|
<ul><li><a href="/wiki/Noncentral_F-distribution" title="Noncentral F-distribution">noncentral</a></li></ul></li>
|
|||
|
|
<li><a href="/wiki/Folded_normal_distribution" title="Folded normal distribution">folded normal</a></li>
|
|||
|
|
<li><a href="/wiki/Fr%C3%A9chet_distribution" title="Fréchet distribution">Fréchet</a></li>
|
|||
|
|
<li><a class="mw-selflink selflink">gamma</a>
|
|||
|
|
<ul><li><a href="/wiki/Generalized_gamma_distribution" title="Generalized gamma distribution">generalized</a></li>
|
|||
|
|
<li><a href="/wiki/Inverse-gamma_distribution" title="Inverse-gamma distribution">inverse</a></li></ul></li>
|
|||
|
|
<li><a href="/wiki/Gamma/Gompertz_distribution" title="Gamma/Gompertz distribution">gamma/Gompertz</a></li>
|
|||
|
|
<li><a href="/wiki/Gompertz_distribution" title="Gompertz distribution">Gompertz</a>
|
|||
|
|
<ul><li><a href="/wiki/Shifted_Gompertz_distribution" title="Shifted Gompertz distribution">shifted</a></li></ul></li>
|
|||
|
|
<li><a href="/wiki/Half-logistic_distribution" title="Half-logistic distribution">half-logistic</a></li>
|
|||
|
|
<li><a href="/wiki/Half-normal_distribution" title="Half-normal distribution">half-normal</a></li>
|
|||
|
|
<li><a href="/wiki/Hotelling%27s_T-squared_distribution" title="Hotelling's T-squared distribution">Hotelling's <i>T</i>-squared</a></li>
|
|||
|
|
<li><a href="/wiki/Inverse_Gaussian_distribution" title="Inverse Gaussian distribution">inverse Gaussian</a>
|
|||
|
|
<ul><li><a href="/wiki/Generalized_inverse_Gaussian_distribution" title="Generalized inverse Gaussian distribution">generalized</a></li></ul></li>
|
|||
|
|
<li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov</a></li>
|
|||
|
|
<li><a href="/wiki/L%C3%A9vy_distribution" title="Lévy distribution">Lévy</a></li>
|
|||
|
|
<li><a href="/wiki/Log-Cauchy_distribution" title="Log-Cauchy distribution">log-Cauchy</a></li>
|
|||
|
|
<li><a href="/wiki/Log-Laplace_distribution" title="Log-Laplace distribution">log-Laplace</a></li>
|
|||
|
|
<li><a href="/wiki/Log-logistic_distribution" title="Log-logistic distribution">log-logistic</a></li>
|
|||
|
|
<li><a href="/wiki/Log-normal_distribution" title="Log-normal distribution">log-normal</a></li>
|
|||
|
|
<li><a href="/wiki/Log-t_distribution" title="Log-t distribution">log-t</a></li>
|
|||
|
|
<li><a href="/wiki/Lomax_distribution" title="Lomax distribution">Lomax</a></li>
|
|||
|
|
<li><a href="/wiki/Matrix-exponential_distribution" title="Matrix-exponential distribution">matrix-exponential</a></li>
|
|||
|
|
<li><a href="/wiki/Maxwell%E2%80%93Boltzmann_distribution" title="Maxwell–Boltzmann distribution">Maxwell–Boltzmann</a></li>
|
|||
|
|
<li><a href="/wiki/Maxwell%E2%80%93J%C3%BCttner_distribution" title="Maxwell–Jüttner distribution">Maxwell–Jüttner</a></li>
|
|||
|
|
<li><a href="/wiki/Mittag-Leffler_distribution" title="Mittag-Leffler distribution">Mittag-Leffler</a></li>
|
|||
|
|
<li><a href="/wiki/Nakagami_distribution" title="Nakagami distribution">Nakagami</a></li>
|
|||
|
|
<li><a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto</a></li>
|
|||
|
|
<li><a href="/wiki/Phase-type_distribution" title="Phase-type distribution">phase-type</a></li>
|
|||
|
|
<li><a href="/wiki/Poly-Weibull_distribution" title="Poly-Weibull distribution">Poly-Weibull</a></li>
|
|||
|
|
<li><a href="/wiki/Rayleigh_distribution" title="Rayleigh distribution">Rayleigh</a></li>
|
|||
|
|
<li><a href="/wiki/Relativistic_Breit%E2%80%93Wigner_distribution" title="Relativistic Breit–Wigner distribution">relativistic Breit–Wigner</a></li>
|
|||
|
|
<li><a href="/wiki/Rice_distribution" title="Rice distribution">Rice</a></li>
|
|||
|
|
<li><a href="/wiki/Truncated_normal_distribution" title="Truncated normal distribution">truncated normal</a></li>
|
|||
|
|
<li><a href="/wiki/Type-2_Gumbel_distribution" title="Type-2 Gumbel distribution">type-2 Gumbel</a></li>
|
|||
|
|
<li><a href="/wiki/Weibull_distribution" title="Weibull distribution">Weibull</a>
|
|||
|
|
<ul><li><a href="/wiki/Discrete_Weibull_distribution" title="Discrete Weibull distribution">discrete</a></li></ul></li>
|
|||
|
|
<li><a href="/wiki/Wilks%27s_lambda_distribution" title="Wilks's lambda distribution">Wilks's lambda</a></li></ul>
|
|||
|
|
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">supported <br />on the whole <br />real line</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
|
|||
|
|
<ul><li><a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy</a></li>
|
|||
|
|
<li><a href="/wiki/Generalized_normal_distribution#Version_1" title="Generalized normal distribution">exponential power</a></li>
|
|||
|
|
<li><a href="/wiki/Fisher%27s_z-distribution" title="Fisher's z-distribution">Fisher's <i>z</i></a></li>
|
|||
|
|
<li><a href="/wiki/Kaniadakis_Gaussian_distribution" title="Kaniadakis Gaussian distribution">Kaniadakis κ-Gaussian</a></li>
|
|||
|
|
<li><a href="/wiki/Gaussian_q-distribution" title="Gaussian q-distribution">Gaussian <i>q</i></a></li>
|
|||
|
|
<li><a href="/wiki/Generalized_normal_distribution" title="Generalized normal distribution">generalized normal</a></li>
|
|||
|
|
<li><a href="/wiki/Generalised_hyperbolic_distribution" title="Generalised hyperbolic distribution">generalized hyperbolic</a></li>
|
|||
|
|
<li><a href="/wiki/Geometric_stable_distribution" title="Geometric stable distribution">geometric stable</a></li>
|
|||
|
|
<li><a href="/wiki/Gumbel_distribution" title="Gumbel distribution">Gumbel</a></li>
|
|||
|
|
<li><a href="/wiki/Holtsmark_distribution" title="Holtsmark distribution">Holtsmark</a></li>
|
|||
|
|
<li><a href="/wiki/Hyperbolic_secant_distribution" title="Hyperbolic secant distribution">hyperbolic secant</a></li>
|
|||
|
|
<li><a href="/wiki/Johnson%27s_SU-distribution" title="Johnson's SU-distribution">Johnson's <i>S<sub>U</sub></i></a></li>
|
|||
|
|
<li><a href="/wiki/Landau_distribution" title="Landau distribution">Landau</a></li>
|
|||
|
|
<li><a href="/wiki/Laplace_distribution" title="Laplace distribution">Laplace</a>
|
|||
|
|
<ul><li><a href="/wiki/Asymmetric_Laplace_distribution" title="Asymmetric Laplace distribution">asymmetric</a></li></ul></li>
|
|||
|
|
<li><a href="/wiki/Logistic_distribution" title="Logistic distribution">logistic</a></li>
|
|||
|
|
<li><a href="/wiki/Noncentral_t-distribution" title="Noncentral t-distribution">noncentral <i>t</i></a></li>
|
|||
|
|
<li><a href="/wiki/Normal_distribution" title="Normal distribution">normal (Gaussian)</a></li>
|
|||
|
|
<li><a href="/wiki/Normal-inverse_Gaussian_distribution" title="Normal-inverse Gaussian distribution">normal-inverse Gaussian</a></li>
|
|||
|
|
<li><a href="/wiki/Skew_normal_distribution" title="Skew normal distribution">skew normal</a></li>
|
|||
|
|
<li><a href="/wiki/Slash_distribution" title="Slash distribution">slash</a></li>
|
|||
|
|
<li><a href="/wiki/Stable_distribution" title="Stable distribution">stable</a></li>
|
|||
|
|
<li><a href="/wiki/Student%27s_t-distribution" title="Student's t-distribution">Student's <i>t</i></a></li>
|
|||
|
|
<li><a href="/wiki/Tracy%E2%80%93Widom_distribution" title="Tracy–Widom distribution">Tracy–Widom</a></li>
|
|||
|
|
<li><a href="/wiki/Variance-gamma_distribution" title="Variance-gamma distribution">variance-gamma</a></li>
|
|||
|
|
<li><a href="/wiki/Voigt_profile" title="Voigt profile">Voigt</a></li></ul>
|
|||
|
|
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">with support <br />whose type varies</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
|
|||
|
|
<ul><li><a href="/wiki/Generalized_chi-squared_distribution" title="Generalized chi-squared distribution">generalized chi-squared</a></li>
|
|||
|
|
<li><a href="/wiki/Generalized_extreme_value_distribution" title="Generalized extreme value distribution">generalized extreme value</a></li>
|
|||
|
|
<li><a href="/wiki/Generalized_Pareto_distribution" title="Generalized Pareto distribution">generalized Pareto</a></li>
|
|||
|
|
<li><a href="/wiki/Marchenko%E2%80%93Pastur_distribution" title="Marchenko–Pastur distribution">Marchenko–Pastur</a></li>
|
|||
|
|
<li><a href="/wiki/Kaniadakis_Exponential_distribution" class="mw-redirect" title="Kaniadakis Exponential distribution">Kaniadakis <i>κ</i>-exponential</a></li>
|
|||
|
|
<li><a href="/wiki/Kaniadakis_Gamma_distribution" title="Kaniadakis Gamma distribution">Kaniadakis <i>κ</i>-Gamma</a></li>
|
|||
|
|
<li><a href="/wiki/Kaniadakis_Weibull_distribution" title="Kaniadakis Weibull distribution">Kaniadakis <i>κ</i>-Weibull</a></li>
|
|||
|
|
<li><a href="/wiki/Kaniadakis_Logistic_distribution" class="mw-redirect" title="Kaniadakis Logistic distribution">Kaniadakis <i>κ</i>-Logistic</a></li>
|
|||
|
|
<li><a href="/wiki/Kaniadakis_Erlang_distribution" title="Kaniadakis Erlang distribution">Kaniadakis <i>κ</i>-Erlang</a></li>
|
|||
|
|
<li><a href="/wiki/Q-exponential_distribution" title="Q-exponential distribution"><i>q</i>-exponential</a></li>
|
|||
|
|
<li><a href="/wiki/Q-Gaussian_distribution" title="Q-Gaussian distribution"><i>q</i>-Gaussian</a></li>
|
|||
|
|
<li><a href="/wiki/Q-Weibull_distribution" title="Q-Weibull distribution"><i>q</i>-Weibull</a></li>
|
|||
|
|
<li><a href="/wiki/Shifted_log-logistic_distribution" title="Shifted log-logistic distribution">shifted log-logistic</a></li>
|
|||
|
|
<li><a href="/wiki/Tukey_lambda_distribution" title="Tukey lambda distribution">Tukey lambda</a></li></ul>
|
|||
|
|
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mixed <br />univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">continuous-<br />discrete</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
|
|||
|
|
<ul><li><a href="/wiki/Rectified_Gaussian_distribution" title="Rectified Gaussian distribution">Rectified Gaussian</a></li></ul>
|
|||
|
|
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">Multivariate <br />(joint)</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
|
|||
|
|
<ul><li><span class="nobold"><i>Discrete: </i></span></li>
|
|||
|
|
<li><a href="/wiki/Ewens%27s_sampling_formula" title="Ewens's sampling formula">Ewens</a></li>
|
|||
|
|
<li><a href="/wiki/Multinomial_distribution" title="Multinomial distribution">multinomial</a>
|
|||
|
|
<ul><li><a href="/wiki/Dirichlet-multinomial_distribution" title="Dirichlet-multinomial distribution">Dirichlet</a></li>
|
|||
|
|
<li><a href="/wiki/Negative_multinomial_distribution" title="Negative multinomial distribution">negative</a></li></ul></li>
|
|||
|
|
<li><span class="nobold"><i>Continuous: </i></span></li>
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|||
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<li><a href="/wiki/Dirichlet_distribution" title="Dirichlet distribution">Dirichlet</a>
|
|||
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<ul><li><a href="/wiki/Generalized_Dirichlet_distribution" title="Generalized Dirichlet distribution">generalized</a></li></ul></li>
|
|||
|
|
<li><a href="/wiki/Multivariate_Laplace_distribution" title="Multivariate Laplace distribution">multivariate Laplace</a></li>
|
|||
|
|
<li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">multivariate normal</a></li>
|
|||
|
|
<li><a href="/wiki/Multivariate_stable_distribution" title="Multivariate stable distribution">multivariate stable</a></li>
|
|||
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<li><a href="/wiki/Multivariate_t-distribution" title="Multivariate t-distribution">multivariate <i>t</i></a></li>
|
|||
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<li><a href="/wiki/Normal-gamma_distribution" title="Normal-gamma distribution">normal-gamma</a>
|
|||
|
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<ul><li><a href="/wiki/Normal-inverse-gamma_distribution" title="Normal-inverse-gamma distribution">inverse</a></li></ul></li>
|
|||
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<li><span class="nobold"><i><a href="/wiki/Random_matrix" title="Random matrix">Matrix-valued: </a></i></span></li>
|
|||
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<li><a href="/wiki/Lewandowski-Kurowicka-Joe_distribution" title="Lewandowski-Kurowicka-Joe distribution">LKJ</a></li>
|
|||
|
|
<li><a href="/wiki/Matrix_normal_distribution" title="Matrix normal distribution">matrix normal</a></li>
|
|||
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|
<li><a href="/wiki/Matrix_t-distribution" title="Matrix t-distribution">matrix <i>t</i></a></li>
|
|||
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<li><a href="/wiki/Matrix_gamma_distribution" title="Matrix gamma distribution">matrix gamma</a>
|
|||
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<ul><li><a href="/wiki/Inverse_matrix_gamma_distribution" title="Inverse matrix gamma distribution">inverse</a></li></ul></li>
|
|||
|
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<li><a href="/wiki/Wishart_distribution" title="Wishart distribution">Wishart</a>
|
|||
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<ul><li><a href="/wiki/Normal-Wishart_distribution" title="Normal-Wishart distribution">normal</a></li>
|
|||
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<li><a href="/wiki/Inverse-Wishart_distribution" title="Inverse-Wishart distribution">inverse</a></li>
|
|||
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<li><a href="/wiki/Normal-inverse-Wishart_distribution" title="Normal-inverse-Wishart distribution">normal-inverse</a></li>
|
|||
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<li><a href="/wiki/Complex_Wishart_distribution" title="Complex Wishart distribution">complex</a></li></ul></li></ul>
|
|||
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</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Directional_statistics" title="Directional statistics">Directional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
|
|||
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<dl><dt><span class="nobold"><i>Univariate (circular) <a href="/wiki/Directional_statistics" title="Directional statistics">directional</a></i></span></dt>
|
|||
|
|
<dd><a href="/wiki/Circular_uniform_distribution" title="Circular uniform distribution">Circular uniform</a></dd>
|
|||
|
|
<dd><a href="/wiki/Von_Mises_distribution" title="Von Mises distribution">univariate von Mises</a></dd>
|
|||
|
|
<dd><a href="/wiki/Wrapped_normal_distribution" title="Wrapped normal distribution">wrapped normal</a></dd>
|
|||
|
|
<dd><a href="/wiki/Wrapped_Cauchy_distribution" title="Wrapped Cauchy distribution">wrapped Cauchy</a></dd>
|
|||
|
|
<dd><a href="/wiki/Wrapped_exponential_distribution" title="Wrapped exponential distribution">wrapped exponential</a></dd>
|
|||
|
|
<dd><a href="/wiki/Wrapped_asymmetric_Laplace_distribution" title="Wrapped asymmetric Laplace distribution">wrapped asymmetric Laplace</a></dd>
|
|||
|
|
<dd><a href="/wiki/Wrapped_L%C3%A9vy_distribution" title="Wrapped Lévy distribution">wrapped Lévy</a></dd>
|
|||
|
|
<dt><span class="nobold"><i>Bivariate (spherical)</i></span></dt>
|
|||
|
|
<dd><a href="/wiki/Kent_distribution" title="Kent distribution">Kent</a></dd>
|
|||
|
|
<dt><span class="nobold"><i>Bivariate (toroidal)</i></span></dt>
|
|||
|
|
<dd><a href="/wiki/Bivariate_von_Mises_distribution" title="Bivariate von Mises distribution">bivariate von Mises</a></dd>
|
|||
|
|
<dt><span class="nobold"><i>Multivariate</i></span></dt>
|
|||
|
|
<dd><a href="/wiki/Von_Mises%E2%80%93Fisher_distribution" title="Von Mises–Fisher distribution">von Mises–Fisher</a></dd>
|
|||
|
|
<dd><a href="/wiki/Bingham_distribution" title="Bingham distribution">Bingham</a></dd></dl>
|
|||
|
|
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Degenerate_distribution" title="Degenerate distribution">Degenerate</a> <br />and <a href="/wiki/Singular_distribution" title="Singular distribution">singular</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
|
|||
|
|
<dl><dt><span class="nobold"><i>Degenerate</i></span></dt>
|
|||
|
|
<dd><a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></dd>
|
|||
|
|
<dt><span class="nobold"><i>Singular</i></span></dt>
|
|||
|
|
<dd><a href="/wiki/Cantor_distribution" title="Cantor distribution">Cantor</a></dd></dl>
|
|||
|
|
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Families</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
|
|||
|
|
<ul><li><a href="/wiki/Circular_distribution" title="Circular distribution">Circular</a></li>
|
|||
|
|
<li><a href="/wiki/Compound_Poisson_distribution" title="Compound Poisson distribution">compound Poisson</a></li>
|
|||
|
|
<li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">elliptical</a></li>
|
|||
|
|
<li><a href="/wiki/Exponential_family" title="Exponential family">exponential</a></li>
|
|||
|
|
<li><a href="/wiki/Natural_exponential_family" title="Natural exponential family">natural exponential</a></li>
|
|||
|
|
<li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">location–scale</a></li>
|
|||
|
|
<li><a href="/wiki/Maximum_entropy_probability_distribution" title="Maximum entropy probability distribution">maximum entropy</a></li>
|
|||
|
|
<li><a href="/wiki/Mixture_distribution" title="Mixture distribution">mixture</a></li>
|
|||
|
|
<li><a href="/wiki/Pearson_distribution" title="Pearson distribution">Pearson</a></li>
|
|||
|
|
<li><a href="/wiki/Tweedie_distribution" title="Tweedie distribution">Tweedie</a></li>
|
|||
|
|
<li><a href="/wiki/Wrapped_distribution" title="Wrapped distribution">wrapped</a></li></ul>
|
|||
|
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</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>
|
|||
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<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Probability_distributions" title="Category:Probability distributions">Category</a></li>
|
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|
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</div></td></tr></tbody></table></div>
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<div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Continuous_distributions" title="Category:Continuous distributions">Continuous distributions</a></li><li><a href="/wiki/Category:Factorial_and_binomial_topics" title="Category:Factorial and binomial topics">Factorial and binomial topics</a></li><li><a href="/wiki/Category:Conjugate_prior_distributions" title="Category:Conjugate prior distributions">Conjugate prior distributions</a></li><li><a href="/wiki/Category:Exponential_family_distributions" title="Category:Exponential family distributions">Exponential family distributions</a></li><li><a href="/wiki/Category:Infinitely_divisible_probability_distributions" title="Category:Infinitely divisible probability distributions">Infinitely divisible probability distributions</a></li><li><a href="/wiki/Category:Survival_analysis" title="Category:Survival analysis">Survival analysis</a></li><li><a href="/wiki/Category:Gamma_and_related_functions" title="Category:Gamma and related functions">Gamma and related functions</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">CS1 maint: multiple names: authors list</a></li><li><a href="/wiki/Category:All_articles_with_incomplete_citations" title="Category:All articles with incomplete citations">All articles with incomplete citations</a></li><li><a href="/wiki/Category:Articles_with_incomplete_citations_from_November_2012" title="Category:Articles with incomplete citations from November 2012">Articles with incomplete citations from November 2012</a></li><li><a href="/wiki/Category:CS1_errors:_missing_periodical" title="Category:CS1 errors: missing periodical">CS1 errors: missing periodical</a></li><li><a href="/wiki/Category:CS1_maint:_numeric_names:_authors_list" title="Category:CS1 maint: numeric names: authors list">CS1 maint: numeric names: authors list</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Wikipedia_introduction_cleanup_from_November_2022" title="Category:Wikipedia introduction cleanup from November 2022">Wikipedia introduction cleanup from November 2022</a></li><li><a href="/wiki/Category:All_pages_needing_cleanup" title="Category:All pages needing cleanup">All pages needing cleanup</a></li><li><a href="/wiki/Category:Articles_covered_by_WikiProject_Wikify_from_November_2022" title="Category:Articles covered by WikiProject Wikify from November 2022">Articles covered by WikiProject Wikify from November 2022</a></li><li><a href="/wiki/Category:All_articles_covered_by_WikiProject_Wikify" title="Category:All articles covered by WikiProject Wikify">All articles covered by WikiProject Wikify</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_September_2012" title="Category:Articles with unsourced statements from September 2012">Articles with unsourced statements from September 2012</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_May_2019" title="Category:Articles with unsourced statements from May 2019">Articles with unsourced statements from May 2019</a></li></ul></div></div>
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