@stdlib/stats/base/dists/kumaraswamy/kurtosis/README.md

Standard library statistical functions.

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# Kurtosis

> [Kumaraswamy's double bounded][kumaraswamy-distribution] distribution [excess kurtosis][kurtosis].

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<section class="intro">

The [excess kurtosis][kurtosis] for a [Kumaraswamy's double bounded][kumaraswamy-distribution] random variable with first shape parameter `a` and second shape parameter `b` is

<!-- <equation class="equation" label="eq:kumaraswamy_kurtosis" align="center" raw="\operatorname{Kurt}\left( X \right) = \frac{ m_4 - ( 4.0 \cdot m_3 \cdot m_1 ) + ( 6.0 \cdot m_2 \cdot m_1^2 ) - ( 3.0 \cdot m_1^4 ) ) }{\left( m_2 - m_1^2 \right)^2}" alt="Excess kurtosis for a Kumaraswamy's double bounded distribution."> -->

<div class="equation" align="center" data-raw-text="\operatorname{Kurt}\left( X \right) = \frac{ m_4 - ( 4.0 \cdot m_3 \cdot m_1 ) + ( 6.0 \cdot m_2 \cdot m_1^2 ) - ( 3.0 \cdot m_1^4 ) ) }{\left( m_2 - m_1^2 \right)^2}" data-equation="eq:kumaraswamy_kurtosis">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/kumaraswamy/kurtosis/docs/img/equation_kumaraswamy_kurtosis.svg" alt="Excess kurtosis for a Kumaraswamy's double bounded distribution.">
    <br>
</div> -->

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where the raw moments of the distribution are given by

<!-- <equation class="equation" label="eq:kumaraswamy_raw_moments" align="center" raw="m_n = b \, B\left(1+\tfrac{n}{a}, b \right)" alt="Raw moments for a Kumaraswamy's double bounded distribution."> -->

<!-- <div class="equation" align="center" data-raw-text="m_n = b \, B\left(1+\tfrac{n}{a}, b \right)" data-equation="eq:kumaraswamy_raw_moments">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/kumaraswamy/kurtosis/docs/img/equation_kumaraswamy_raw_moments.svg" alt="Raw moments for a Kumaraswamy's double bounded distribution.">
    <br>
</div>

<!-- </equation> -->

with `B` denoting the [beta function][beta-function].

</section>

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<section class="usage">

## Usage

```javascript
var kurtosis = require( '@stdlib/stats/base/dists/kumaraswamy/kurtosis' );
```

#### kurtosis( a, b )

Returns the [excess kurtosis][kurtosis] of a [Kumaraswamy's double bounded][kumaraswamy-distribution] distribution with first shape parameter `a` and second shape parameter `b`.

```javascript
var v = kurtosis( 1.0, 1.0 );
// returns ~1.8

v = kurtosis( 4.0, 12.0 );
// returns ~2.704

v = kurtosis( 2.0, 8.0 );
// returns ~2.666
```

If provided `NaN` as any argument, the function returns `NaN`.

```javascript
var v = kurtosis( NaN, 2.0 );
// returns NaN

v = kurtosis( 2.0, NaN );
// returns NaN
```

If provided `a <= 0`, the function returns `NaN`.

```javascript
var y = kurtosis( -1.0, 0.5 );
// returns NaN

y = kurtosis( 0.0, 0.5 );
// returns NaN
```

If provided `b <= 0`, the function returns `NaN`.

```javascript
var y = kurtosis( 0.5, -1.0 );
// returns NaN

y = kurtosis( 0.5, 0.0 );
// returns NaN
```

</section>

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</section>

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<section class="examples">

## Examples

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

```javascript
var randu = require( '@stdlib/random/base/randu' );
var kurtosis = require( '@stdlib/stats/base/dists/kumaraswamy/kurtosis' );

var a;
var b;
var v;
var i;

for ( i = 0; i < 10; i++ ) {
    a = randu() * 10.0;
    b = randu() * 10.0;
    v = kurtosis( a, b );
    console.log( 'a: %d, b: %d, Kurt(X;a,b): %d', a.toFixed( 4 ), b.toFixed( 4 ), v.toFixed( 4 ) );
}
```

</section>

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<section class="links">

[beta-function]: https://en.wikipedia.org/wiki/Beta_function

[kumaraswamy-distribution]: https://en.wikipedia.org/wiki/Kumaraswamy_distribution

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

</section>

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