README.md

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Kurtosis

Fréchet distribution kurtosis.

The excess kurtosis for a Fréchet random variable shape α > 0, scale s > 0, and location parameter m is

$$\mathop{\mathrm{Kurt}} = \begin{cases} -6+{\frac{\Gamma \left(1-{\frac{4}{\alpha }}\right)-4\Gamma \left(1-{\frac{3}{\alpha }}\right)\Gamma \left(1-{\frac{1}{\alpha }}\right)+3\Gamma^{2}\left(1-{\frac{2}{\alpha }}\right)}{\left[\Gamma \left(1-{\frac{2}{\alpha }}\right)-\Gamma^{2}\left(1-{\frac{1}{\alpha }}\right)\right]^{2}}} & {\text{ for }}\alpha >4\\\ \infty & \text{ otherwise }\end{cases}$$

where Γ is the gamma function.

Usage

import kurtosis from 'https://cdn.jsdelivr.net/gh/stdlib-js/stats-base-dists-frechet-kurtosis@deno/mod.js';

kurtosis( alpha, s, m )

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

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.

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

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

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.

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.

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

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

Examples

import randu from 'https://cdn.jsdelivr.net/gh/stdlib-js/random-base-randu@deno/mod.js';
import EPS from 'https://cdn.jsdelivr.net/gh/stdlib-js/constants-float64-eps@deno/mod.js';
import kurtosis from 'https://cdn.jsdelivr.net/gh/stdlib-js/stats-base-dists-frechet-kurtosis@deno/mod.js';

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()*20.0 ) - 40.0;
    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 ) );
}

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Notice

This package is part of stdlib, a standard library 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, see the main project repository.

Community

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License

See LICENSE.

Copyright

Copyright © 2016-2024. The Stdlib Authors.