npm package: @stdlib/stats-base-dists-pareto-type1-kurtosis
Description: Pareto (Type I) distribution excess kurtosis.
Author: The Stdlib Authors
Latest version: 0.2.3

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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] -->

> [Pareto (Type I)][pareto-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 [Pareto (Type I)][pareto-distribution] random variable with shape parameter `α` and scale parameter `β` is

<!-- <equation class="equation" label="eq:pareto_type1_kurtosis" align="center" raw="\operatorname{Kurt}\left( X \right) = \frac{6(\alpha^3+\alpha^2-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)}" alt="Excess kurtosis for a Pareto (Type I) distribution."> -->

<div class="equation" align="center" data-raw-text="\operatorname{Kurt}\left( X \right) = \frac{6(\alpha^3+\alpha^2-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)}" data-equation="eq:pareto_type1_kurtosis">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/pareto-type1/kurtosis/docs/img/equation_pareto_type1_kurtosis.svg" alt="Excess kurtosis for a Pareto (Type I) distribution.">
    <br>
</div>

<!-- </equation> -->

for `α > 4` and `β > 0`. Otherwise, the kurtosis is not defined.

</section>

<!-- /.intro -->

<!-- Package usage documentation. -->

<section class="installation">

## Installation

```bash
npm install @stdlib/stats-base-dists-pareto-type1-kurtosis
```

</section>

<section class="usage">

## Usage

```javascript
var kurtosis = require( '@stdlib/stats-base-dists-pareto-type1-kurtosis' );
```

#### kurtosis( alpha, beta )

Returns the [excess kurtosis][kurtosis] of a [Pareto (Type I)][pareto-distribution] distribution with parameters `alpha` (shape parameter) and `beta` (scale parameter).

```javascript
var v = kurtosis( 5.0, 1.0 );
// returns ~70.8

v = kurtosis( 4.5, 12.0 );
// returns ~146.444

v = kurtosis( 8.0, 2.0 );
// returns ~19.725
```

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

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

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

v = kurtosis( 4.0, 2.0 );
// returns NaN

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

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

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

v = kurtosis( 1.0, -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 EPS = require( '@stdlib/constants-float64-eps' );
var logEachMap = require( '@stdlib/console-log-each-map' );
var kurtosis = require( '@stdlib/stats-base-dists-pareto-type1-kurtosis' );

var opts = {
    'dtype': 'float64'
};
var alpha = uniform( 10, EPS + 4.0, 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/pareto-type1/kurtosis.h"
```

#### stdlib_base_dists_pareto_type1_kurtosis( alpha, beta )

Returns the excess kurtosis of a Pareto (Type I) distribution.

```c
double out = stdlib_base_dists_pareto_type1_kurtosis( 5.0, 1.0 );
// returns ~70.8
```

The function accepts the following arguments:

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

```c
double stdlib_base_dists_pareto_type1_kurtosis( const double alpha, const double beta );
```

</section>

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<section class="notes">

</section>

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

<section class="examples">

### Examples

```c
#include "stdlib/stats/base/dists/pareto-type1/kurtosis.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( 0.0, 10.0 ) + 4.0;
        beta = random_uniform( 0.0, 10.0 );
        y = stdlib_base_dists_pareto_type1_kurtosis( alpha, beta );
        printf( "α: %lf, β: %lf, kurtosis(X;α,β): %lf\n", alpha, beta, y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

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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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[pareto-distribution]: https://en.wikipedia.org/wiki/Pareto_distribution

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

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