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Fréchet distribution kurtosis.

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Apache-2.0 license

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About stdlib...

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Kurtosis

Fréchet distributionkurtosis.

Theexcess kurtosis for aFréchet random variable shapeα > 0, scales > 0, and location parameterm is

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

whereΓ is thegamma function.

Installation

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

Alternatively,

To load the package in a website via ascript tag without installation and bundlers, use theES Module available on theesm branch (seeREADME).
If you are using Deno, visit thedeno branch (seeREADME for usage intructions).
For use in Observable, or in browser/node environments, use theUniversal Module Definition (UMD) build available on theumd branch (seeREADME).

Thebranches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

Usage

varkurtosis=require('@stdlib/stats-base-dists-frechet-kurtosis');

kurtosis( alpha, s, m )

Returns theexcess kurtosis for aFréchet distribution with shapealpha > 0, scales > 0, and location parameterm.

vary=kurtosis(5.0,1.0,1.0);// returns ~45.092y=kurtosis(5.0,10.0,1.0);// returns ~45.092y=kurtosis(5.0,1.0,2.0);// returns ~45.092

If0 < alpha <= 4, the function returns+Infinity.

vary=kurtosis(2.5,1.0,1.0);// returns Infinity

If providedNaN as any argument, the function returnsNaN.

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

If providedalpha <= 0, the function returnsNaN.

vary=kurtosis(0.0,3.0,2.0);// returns NaNy=kurtosis(0.0,-1.0,2.0);// returns NaN

If provideds <= 0, the function returnsNaN.

vary=kurtosis(1.0,0.0,2.0);// returns NaNy=kurtosis(1.0,-1.0,2.0);// returns NaN

Examples

varrandu=require('@stdlib/random-base-randu');varEPS=require('@stdlib/constants-float64-eps');varkurtosis=require('@stdlib/stats-base-dists-frechet-kurtosis');varalpha;varm;vars;vary;vari;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));}

C APIs

Usage

#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 shapealpha, scales, and locationm.

doubley=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.

doublestdlib_base_dists_frechet_kurtosis(constdoublealpha,constdoubles,constdoublem );

Examples

#include"stdlib/stats/base/dists/frechet/kurtosis.h"#include"stdlib/constants/float64/eps.h"#include<stdlib.h>#include<stdio.h>staticdoublerandom_uniform(constdoublemin,constdoublemax ) {doublev= (double)rand() / ( (double)RAND_MAX+1.0 );returnmin+ (v*(max-min) );}intmain(void ) {doublealpha;doubles;doublem;doubley;inti;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 );    }}

Notice

This package is part ofstdlib, 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 developstdlib, see the main projectrepository.