npm package: @stdlib/stats-base-dists-frechet-kurtosis
Description: Fréchet distribution kurtosis.
Author: The Stdlib Authors
Latest version: 0.3.1

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# Kurtosis

[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] <!-- [![dependencies][dependencies-image]][dependencies-url] -->

> [Fréchet][frechet-distribution] distribution [kurtosis][kurtosis].

<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->

<section class="intro">

The [excess kurtosis][kurtosis] for a [Fréchet][frechet-distribution] random variable shape `α > 0`, scale `s > 0`, and location parameter `m` is

<!-- <equation class="equation" label="eq:frechet_kurtosis" align="center" raw="\operatorname{Kurt} = \begin{cases} -6+{\frac{\Gamma \left(1-{\frac{4}{\alpha }}\right)-4\Gamma \left(1-{\frac{3}{\alpha }}\right)\Gamma \left(1-{\frac{1}{\alpha }}\right)+3\Gamma^{2}\left(1-{\frac{2}{\alpha }}\right)}{\left[\Gamma \left(1-{\frac{2}{\alpha }}\right)-\Gamma^{2}\left(1-{\frac{1}{\alpha }}\right)\right]^{2}}} & {\text{ for }}\alpha >4\\\ \infty & \text{ otherwise }\end{cases}" alt="Kurtosis for a Fréchet distribution."> -->

<div class="equation" align="center" data-raw-text="\operatorname{Kurt} = \begin{cases} -6+{\frac{\Gamma \left(1-{\frac{4}{\alpha }}\right)-4\Gamma \left(1-{\frac{3}{\alpha }}\right)\Gamma \left(1-{\frac{1}{\alpha }}\right)+3\Gamma^{2}\left(1-{\frac{2}{\alpha }}\right)}{\left[\Gamma \left(1-{\frac{2}{\alpha }}\right)-\Gamma^{2}\left(1-{\frac{1}{\alpha }}\right)\right]^{2}}} &amp; {\text{ for }}\alpha &gt;4\\\ \infty &amp; \text{ otherwise }\end{cases}" data-equation="eq:frechet_kurtosis">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/frechet/kurtosis/docs/img/equation_frechet_kurtosis.svg" alt="Kurtosis for a Fréchet distribution.">
    <br>
</div>

<!-- </equation> -->

where `Γ` is the [gamma function][gamma-function].

</section>

<!-- /.intro -->

<!-- Package usage documentation. -->

<section class="installation">

## Installation

```bash
npm install @stdlib/stats-base-dists-frechet-kurtosis
```

</section>

<section class="usage">

## Usage

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

#### kurtosis( alpha, s, m )

Returns the [excess kurtosis][kurtosis] for a [Fréchet][frechet-distribution] distribution with shape `alpha > 0`, scale `s > 0`, and location parameter `m`.

```javascript
var y = kurtosis( 5.0, 1.0, 1.0 );
// returns ~45.092

y = kurtosis( 5.0, 10.0, 1.0 );
// returns ~45.092

y = kurtosis( 5.0, 1.0, 2.0 );
// returns ~45.092
```

If `0 < alpha <= 4`, the function returns `+Infinity`.

```javascript
var y = kurtosis( 2.5, 1.0, 1.0 );
// returns Infinity
```

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

```javascript
var y = kurtosis( NaN, 1.0, -2.0 );
// returns NaN

y = kurtosis( 1.0, NaN, -2.0 );
// returns NaN

y = kurtosis( 1.0, 1.0, NaN );
// returns NaN
```

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

```javascript
var y = kurtosis( 0.0, 3.0, 2.0 );
// returns NaN

y = kurtosis( 0.0, -1.0, 2.0 );
// returns NaN
```

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

```javascript
var y = kurtosis( 1.0, 0.0, 2.0 );
// returns NaN

y = kurtosis( 1.0, -1.0, 2.0 );
// returns NaN
```

</section>

<!-- /.usage -->

<!-- Package usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="notes">

</section>

<!-- /.notes -->

<!-- Package usage examples. -->

<section class="examples">

## Examples

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

```javascript
var randu = require( '@stdlib/random-base-randu' );
var EPS = require( '@stdlib/constants-float64-eps' );
var kurtosis = require( '@stdlib/stats-base-dists-frechet-kurtosis' );

var alpha;
var m;
var s;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    alpha = ( randu()*20.0 ) + EPS;
    s = ( randu()*20.0 ) + EPS;
    m = ( randu()*40.0 ) - 20.0;
    y = kurtosis( alpha, s, m );
    console.log( 'α: %d, s: %d, m: %d, Kurt(X;α,s,m): %d', alpha.toFixed( 4 ), s.toFixed( 4 ), m.toFixed( 4 ), y.toFixed( 4 ) );
}
```

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->

<section class="intro">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

### Usage

```c
#include "stdlib/stats/base/dists/frechet/kurtosis.h"
```

#### stdlib_base_dists_frechet_kurtosis( alpha, s, m )

Returns the excess kurtosis for a Fréchet distribution with shape `alpha`, scale `s`, and location `m`.

```c
double y = stdlib_base_dists_frechet_kurtosis( 5.0, 2.0, 0.0 );
// returns ~45.092
```

The function accepts the following arguments:

-   **alpha**: `[in] double` shape parameter.
-   **s**: `[in] double` scale parameter.
-   **m**: `[in] double` location parameter.

```c
double stdlib_base_dists_frechet_kurtosis( const double alpha, const double s, const double m );
```

</section>

<!-- /.usage -->

<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="notes">

</section>

<!-- /.notes -->

<!-- C API usage examples. -->

<section class="examples">

### Examples

```c
#include "stdlib/stats/base/dists/frechet/kurtosis.h"
#include "stdlib/constants/float64/eps.h"
#include <stdlib.h>
#include <stdio.h>

static double random_uniform( const double min, const double max ) {
    double v = (double)rand() / ( (double)RAND_MAX + 1.0 );
    return min + ( v*(max-min) );
}

int main( void ) {
   double alpha;
   double s;
   double m;
   double y;
   int i;

    for ( i = 0; i < 10; i++ ) {
       alpha = random_uniform( STDLIB_CONSTANT_FLOAT64_EPS, 20.0 );
       s = random_uniform( STDLIB_CONSTANT_FLOAT64_EPS, 20.0 );
       m = random_uniform( -40.0, -20.0 );
       y = stdlib_base_dists_frechet_kurtosis( alpha, s, m );
       printf( "α: %.4lf, s: %.4lf, m: %.4lf, Kurt(X;α,s,m): %.4lf\n", alpha, s, m, y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

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<section class="related">

</section>

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* * *

## Notice

This package is part of [stdlib][stdlib], a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop [stdlib][stdlib], see the main project [repository][stdlib].

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## License

See [LICENSE][stdlib-license].


## Copyright

Copyright &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].

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[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/stats-base-dists-frechet-kurtosis/main/LICENSE

[frechet-distribution]: https://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution

[gamma-function]: https://en.wikipedia.org/wiki/Gamma_function

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

</section>

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