Kurtosis

Kumaraswamy's double bounded distribution excess kurtosis.

The excess kurtosis for a Kumaraswamy's double bounded random variable with first shape parameter a and second shape parameter b is

upper K u r t left-parenthesis upper X right-parenthesis equals StartFraction m 4 minus left-parenthesis 4.0 dot m 3 dot m 1 right-parenthesis plus left-parenthesis 6.0 dot m 2 dot m 1 squared right-parenthesis minus left-parenthesis 3.0 dot m 1 Superscript 4 Baseline right-parenthesis right-parenthesis Over left-parenthesis m 2 minus m 1 squared right-parenthesis squared EndFraction

where the raw moments of the distribution are given by

m Subscript n Baseline equals b upper B left-parenthesis 1 plus StartFraction n Over a EndFraction comma b right-parenthesis

with B denoting the beta function.

Usage

var kurtosis = require( '@stdlib/math/base/dists/kumaraswamy/kurtosis' );

kurtosis( a, b )

Returns the excess kurtosis of a Kumaraswamy's double bounded distribution with first shape parameter a and second shape parameter b.

var v = kurtosis( 1.0, 1.0 );
// returns ~1.8

v = kurtosis( 4.0, 12.0 );
// returns ~2.704

v = kurtosis( 2.0, 8.0 );
// returns ~2.666

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

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

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

If provided a <= 0, the function returns NaN.

var y = kurtosis( -1.0, 0.5 );
// returns NaN

y = kurtosis( 0.0, 0.5 );
// returns NaN

If provided b <= 0, the function returns NaN.

var y = kurtosis( 0.5, -1.0 );
// returns NaN

y = kurtosis( 0.5, 0.0 );
// returns NaN

Examples

var randu = require( '@stdlib/random/base/randu' );
var kurtosis = require( '@stdlib/math/base/dists/kumaraswamy/kurtosis' );

var a;
var b;
var v;
var i;

for ( i = 0; i < 10; i++ ) {
    a = randu() * 10.0;
    b = randu() * 10.0;
    v = kurtosis( a, b );
    console.log( 'a: %d, b: %d, Kurt(X;a,b): %d', a.toFixed( 4 ), b.toFixed( 4 ), v.toFixed( 4 ) );
}