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# incrkurtosis

> Compute a [corrected sample excess kurtosis][sample-excess-kurtosis] incrementally.

<section class="intro">

The [kurtosis][sample-excess-kurtosis] for a random variable `X` is defined as

<!-- <equation class="equation" label="eq:kurtosis" align="center" raw="\operatorname{Kurtosis}[X] = \mathrm{E}\biggl[ \biggl( \frac{X - \mu}{\sigma} \biggr)^4 \biggr]" alt="Equation for the kurtosis."> -->

<div class="equation" align="center" data-raw-text="\operatorname{Kurtosis}[X] = \mathrm{E}\biggl[ \biggl( \frac{X - \mu}{\sigma} \biggr)^4 \biggr]" data-equation="eq:kurtosis">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/27e2a43c70db648bb5bbc3fd0cdee050c25adc0b/lib/node_modules/@stdlib/stats/incr/kurtosis/docs/img/equation_kurtosis.svg" alt="Equation for the kurtosis.">
    <br>
</div>

<!-- </equation> -->

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

<!-- <equation class="equation" label="eq:sample_excess_kurtosis" align="center" raw="g_2 = \frac{m_4}{m_2^2} - 3 = \frac{\frac{1}{n} \sum_{i=0}^{n-1} (x_i - \bar{x})^4}{\biggl(\frac{1}{n} \sum_{i=0}^{n-1} (x_i - \bar{x})^2\biggr)^2}" alt="Equation for the sample excess kurtosis."> -->

<div class="equation" align="center" data-raw-text="g_2 = \frac{m_4}{m_2^2} - 3 = \frac{\frac{1}{n} \sum_{i=0}^{n-1} (x_i - \bar{x})^4}{\biggl(\frac{1}{n} \sum_{i=0}^{n-1} (x_i - \bar{x})^2\biggr)^2}" data-equation="eq:sample_excess_kurtosis">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/27e2a43c70db648bb5bbc3fd0cdee050c25adc0b/lib/node_modules/@stdlib/stats/incr/kurtosis/docs/img/equation_sample_excess_kurtosis.svg" alt="Equation for the sample excess kurtosis.">
    <br>
</div>

<!-- </equation> -->

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

<!-- <equation class="equation" label="eq:corrected_sample_excess_kurtosis" align="center" raw="G_2 = \frac{(n+1)n}{(n-1)(n-2)(n-3)} \frac{\sum_{i=0}^{n-1} (x_i - \bar{x})^4}{\biggl(\sum_{i=0}^{n-1} (x_i - \bar{x})^2\biggr)^2} - 3 \frac{(n-1)^2}{(n-2)(n-3)}" alt="Equation for the corrected sample excess kurtosis."> -->

<div class="equation" align="center" data-raw-text="G_2 = \frac{(n+1)n}{(n-1)(n-2)(n-3)} \frac{\sum_{i=0}^{n-1} (x_i - \bar{x})^4}{\biggl(\sum_{i=0}^{n-1} (x_i - \bar{x})^2\biggr)^2} - 3 \frac{(n-1)^2}{(n-2)(n-3)}" data-equation="eq:corrected_sample_excess_kurtosis">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/27e2a43c70db648bb5bbc3fd0cdee050c25adc0b/lib/node_modules/@stdlib/stats/incr/kurtosis/docs/img/equation_corrected_sample_excess_kurtosis.svg" alt="Equation for the corrected sample excess kurtosis.">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<section class="usage">

## 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
```

</section>

<!-- /.usage -->

<section class="notes">

## 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.

</section>

<!-- /.notes -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```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() );
```

</section>

<!-- /.examples -->

* * *

<section class="references">

## 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].

</section>

<!-- /.references -->

<section class="links">

[sample-excess-kurtosis]: https://en.wikipedia.org/wiki/Kurtosis

[@joanes:1998]: http://onlinelibrary.wiley.com/doi/10.1111/1467-9884.00122/

</section>

<!-- /.links -->
