# incrkurtosis
> Compute a [corrected sample excess kurtosis][sample-excess-kurtosis] incrementally.
The [kurtosis][sample-excess-kurtosis] for a random variable `X` is defined as
Using a univariate normal distribution as the standard of comparison, the [excess kurtosis][sample-excess-kurtosis] is the kurtosis minus `3`.
For a sample of `n` values, the [sample excess kurtosis][sample-excess-kurtosis] is
where `m_4` is the sample fourth central moment and `m_2` is the sample second central moment.
The previous equation is, however, a biased estimator of the population excess kurtosis. An alternative estimator which is unbiased under normality is
## Usage
```javascript
var incrkurtosis = require( '@stdlib/stats/incr/kurtosis' );
```
#### incrkurtosis()
Returns an accumulator `function` which incrementally computes a [corrected sample excess kurtosis][sample-excess-kurtosis].
```javascript
var accumulator = incrkurtosis();
```
#### accumulator( \[x] )
If provided an input value `x`, the accumulator function returns an updated [corrected sample excess kurtosis][sample-excess-kurtosis]. If not provided an input value `x`, the accumulator function returns the current [corrected sample excess kurtosis][sample-excess-kurtosis].
```javascript
var accumulator = incrkurtosis();
var kurtosis = accumulator( 2.0 );
// returns null
kurtosis = accumulator( 2.0 );
// returns null
kurtosis = accumulator( -4.0 );
// returns null
kurtosis = accumulator( -4.0 );
// returns -6.0
```
## Notes
- Input values are **not** type checked. If provided `NaN` or a value which, when used in computations, results in `NaN`, the accumulated value is `NaN` for **all** future invocations. If non-numeric inputs are possible, you are advised to type check and handle accordingly **before** passing the value to the accumulator function.
## Examples
```javascript
var randu = require( '@stdlib/random/base/randu' );
var incrkurtosis = require( '@stdlib/stats/incr/kurtosis' );
var accumulator;
var v;
var i;
// Initialize an accumulator:
accumulator = incrkurtosis();
// For each simulated datum, update the corrected sample excess kurtosis...
for ( i = 0; i < 100; i++ ) {
v = randu() * 100.0;
accumulator( v );
}
console.log( accumulator() );
```
* * *
## References
- Joanes, D. N., and C. A. Gill. 1998. "Comparing measures of sample skewness and kurtosis." _Journal of the Royal Statistical Society: Series D (The Statistician)_ 47 (1). Blackwell Publishers Ltd: 183–89. doi:[10.1111/1467-9884.00122][@joanes:1998].
[sample-excess-kurtosis]: https://en.wikipedia.org/wiki/Kurtosis
[@joanes:1998]: http://onlinelibrary.wiley.com/doi/10.1111/1467-9884.00122/
