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

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# Kurtosis

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> [Inverse gamma][invgamma-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 an [inverse gamma][invgamma-distribution] random variable with shape parameter `α` and rate parameter `β` is

<!-- <equation class="equation" label="eq:invgamma_kurtosis" align="center" raw="\operatorname{Kurt}\left( X \right) = \frac {30\,\alpha -66}{(\alpha -3)(\alpha -4)}" alt="Excess kurtosis for an inverse gamma distribution."> -->

<div class="equation" align="center" data-raw-text="\operatorname{Kurt}\left( X \right) = \frac {30\,\alpha -66}{(\alpha -3)(\alpha -4)}" data-equation="eq:invgamma_kurtosis">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/invgamma/kurtosis/docs/img/equation_invgamma_kurtosis.svg" alt="Excess kurtosis for an inverse gamma distribution.">
    <br>
</div>

<!-- </equation> -->

when `α > 4`. Otherwise, the kurtosis is not defined.

</section>

<!-- /.intro -->

<!-- Package usage documentation. -->

<section class="installation">

## Installation

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

</section>

<section class="usage">

## Usage

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

#### kurtosis( alpha, beta )

Returns the [excess kurtosis][kurtosis] of an [inverse gamma][invgamma-distribution] distribution with parameters `alpha` (shape parameter) and `beta` (rate parameter).

```javascript
var v = kurtosis( 7.0, 5.0 );
// returns 12.0

v = kurtosis( 6.0, 12.0 );
// returns 19.0

v = kurtosis( 8.0, 2.0 );
// returns ~8.7
```

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

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

v = kurtosis( 5.0, NaN );
// returns NaN
```

If provided `alpha <= 4`, the function returns `NaN`.

```javascript
var v = kurtosis( 3.0, 1.0 );
// returns NaN

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

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

```javascript
var v = kurtosis( 5.0, 0.0 );
// returns NaN

v = kurtosis( 5.0, -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 EPS = require( '@stdlib/constants-float64-eps' );
var uniform = require( '@stdlib/random-array-uniform' );
var logEachMap = require( '@stdlib/console-log-each-map' );
var kurtosis = require( '@stdlib/stats-base-dists-invgamma-kurtosis' );

var opts = {
    'dtype': 'float64'
};
var alpha = uniform( 10, EPS, 10.0, opts );
var beta = uniform( 10, EPS, 10.0, opts );

logEachMap( 'α: %0.4f, β: %0.4f, Kurt(X;α,β): %0.4f', alpha, beta, 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/invgamma/kurtosis.h"
```

#### stdlib_base_dists_invgamma_kurtosis( alpha, beta )

Evaluates the [excess kurtosis][kurtosis] of an [inverse gamma][invgamma-distribution] distribution with parameters `alpha` (shape parameter) and `beta` (rate parameter).

```c
double out = stdlib_base_dists_invgamma_kurtosis( 6.0, 1.0 );
// returns 19.0
```

The function accepts the following arguments:

-   **alpha**: `[in] double` shape parameter.
-   **beta**: `[in] double` rate parameter.

```c
double stdlib_base_dists_invgamma_kurtosis( const double alpha, const double beta );
```

</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/invgamma/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 beta;
    double y;
    int i;

    for ( i = 0; i < 25; i++ ) {
        alpha = random_uniform( STDLIB_CONSTANT_FLOAT64_EPS, 10.0 );
        beta = random_uniform( STDLIB_CONSTANT_FLOAT64_EPS, 10.0 );
        y = stdlib_base_dists_invgamma_kurtosis( alpha, beta );
        printf( "α: %lf, β: %lf, Kurt(X;α,β): %lf\n", alpha, beta, y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

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</section>

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<section class="related">

</section>

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<section class="main-repo" >

* * *

## 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-invgamma-kurtosis/main/LICENSE

[invgamma-distribution]: https://en.wikipedia.org/wiki/Inverse-gamma_distribution

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

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