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

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# Kurtosis

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> [F][f-distribution] distribution [excess 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 [F][f-distribution] random variable with numerator degrees of freedom `d1` and denominator degrees of freedom `d2` is

<!-- <equation class="equation" label="eq:f_kurtosis" align="center" raw="\operatorname{Kurt}\left( X \right) = \gamma_{2}=12{\frac{d_{1}(5d_{2}-22)(d_{1}+d_{2}-2)+(d_{2}-4)(d_{2}-2)^{2}}{d_{1}(d_{2}-6)(d_{2}-8)(d_{1}+d_{2}-2)}}." alt="Excess kurtosis for an F distribution."> -->

<div class="equation" align="center" data-raw-text="\operatorname{Kurt}\left( X \right) = \gamma_{2}=12{\frac{d_{1}(5d_{2}-22)(d_{1}+d_{2}-2)+(d_{2}-4)(d_{2}-2)^{2}}{d_{1}(d_{2}-6)(d_{2}-8)(d_{1}+d_{2}-2)}}." data-equation="eq:f_kurtosis">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/f/kurtosis/docs/img/equation_f_kurtosis.svg" alt="Excess kurtosis for an F distribution.">
    <br>
</div>

<!-- </equation> -->

for `d1 > 0` and `d2 > 8`. Otherwise, the kurtosis is not defined.

</section>

<!-- /.intro -->

<!-- Package usage documentation. -->

<section class="installation">

## Installation

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

</section>

<section class="usage">

## Usage

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

#### kurtosis( d1, d2 )

Returns the [excess kurtosis][kurtosis] of an [F][f-distribution] distribution with parameters `d1` (numerator degrees of freedom) and `d2` (denominator degrees of freedom).

```javascript
var v = kurtosis( 4.0, 9.0 );
// returns ~114.273

v = kurtosis( 4.0, 12.0 );
// returns ~26.143

v = kurtosis( 8.0, 9.0 );
// returns ~100.167
```

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

```javascript
var v = kurtosis( NaN, 7.0 );
// returns NaN

v = kurtosis( 3.0, NaN );
// returns NaN
```

If provided `d1 <= 0`, the function returns `NaN`.

```javascript
var v = kurtosis( 0.0, 9.0 );
// returns NaN

v = kurtosis( -1.0, 9.0 );
// returns NaN
```

If provided `d2 <= 8`, the function returns `NaN`.

```javascript
var v = kurtosis( 1.0, 8.0 );
// returns NaN

v = kurtosis( 2.2, 5.5 );
// returns NaN

v = kurtosis( 3.0, -1.0 );
// 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 uniform = require( '@stdlib/random-array-uniform' );
var logEachMap = require( '@stdlib/console-log-each-map' );
var EPS = require( '@stdlib/constants-float64-eps' );
var kurtosis = require( '@stdlib/stats-base-dists-f-kurtosis' );

var opts = {
    'dtype': 'float64'
};
var d1 = uniform( 10, EPS, 10.0, opts );
var d2 = uniform( 10, EPS, 20.0, opts );

logEachMap( 'd1: %0.4f, d2: %0.4f, Kurt(X;d1,d2): %0.4f', d1, d2, kurtosis );
```

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

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<section class="intro">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

### Usage

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

#### stdlib_base_dists_f_kurtosis( d1, d2 )

Evaluates the [excess kurtosis][kurtosis] of an [F][f-distribution] distribution with parameters `d1` (numerator degrees of freedom) and `d2` (denominator degrees of freedom).

```c
double out = stdlib_base_dists_f_kurtosis( 3.0, 9.0 );
// returns ~124.667
```

The function accepts the following arguments:

-   **d1**: `[in] double` numerator degrees of freedom.
-   **d2**: `[in] double` denominator degrees of freedom.

```c
double stdlib_base_dists_f_kurtosis( const double d1, const double d2 );
```

</section>

<!-- /.usage -->

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<section class="notes">

</section>

<!-- /.notes -->

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

<section class="examples">

### Examples

```c
#include "stdlib/stats/base/dists/f/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 d1;
    double d2;
    double y;
    int i;

    for ( i = 0; i < 25; i++ ) {
        d1 = random_uniform( STDLIB_CONSTANT_FLOAT64_EPS, 10.0 );
        d1 = random_uniform( STDLIB_CONSTANT_FLOAT64_EPS, 10.0 );
        y = stdlib_base_dists_f_kurtosis( d1, d2 );
        printf( "d1: %lf, d2: %lf, Kurt(X;d1,d2): %lf\n", d1, d2, y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

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* * *

## Notice

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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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[f-distribution]: https://en.wikipedia.org/wiki/F_distribution

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

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