Kurtosis

Beta prime distribution excess kurtosis.

The excess kurtosis for a beta prime random variable with first shape parameter α and second shape parameter β is

upper K u r t left-parenthesis upper X right-parenthesis equals 6 StartFraction alpha left-parenthesis alpha plus beta minus 1 right-parenthesis left-parenthesis 5 beta minus 11 right-parenthesis plus left-parenthesis beta minus 1 right-parenthesis squared left-parenthesis beta minus 2 right-parenthesis Over alpha left-parenthesis alpha plus beta minus 1 right-parenthesis left-parenthesis beta minus 3 right-parenthesis left-parenthesis beta minus 4 right-parenthesis EndFraction

when α > 0 and β > 4. Otherwise, the excess kurtosis is not defined.

Usage

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

kurtosis( alpha, beta )

Returns the excess kurtosis of a beta prime distribution with parameters alpha (first shape parameter) and beta (second shape parameter).

var v = kurtosis( 2.0, 5.0 );
// returns 54.0

v = kurtosis( 4.0, 12.0 );
// returns ~5.764

v = kurtosis( 12.0, 6.0 );
// returns ~19.49

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

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

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

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

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

v = kurtosis( -1.0, 5.0 );
// returns NaN

If provided beta <= 4, the function returns NaN.

var v = kurtosis( 1.0, 3.5 );
// returns NaN

v = kurtosis( 1.0, 2.0 );
// returns NaN

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

Examples

var randu = require( '@stdlib/random/base/randu' );
var EPS = require( '@stdlib/constants/math/float64-eps' );
var kurtosis = require( '@stdlib/math/base/dists/betaprime/kurtosis' );

var alpha;
var beta;
var v;
var i;

for ( i = 0; i < 10; i++ ) {
    alpha = ( randu()*10.0 ) + EPS;
    beta = ( randu()*10.0 ) + 4.0 + EPS;
    v = kurtosis( alpha, beta );
    console.log( 'α: %d, β: %d, Kurt(X;α,β): %d', alpha.toFixed( 4 ), beta.toFixed( 4 ), v.toFixed( 4 ) );
}