Kurtosis

Weibull distribution excess kurtosis.

The excess kurtosis for a Weibull random variable with shape parameter λ > 0 and scale parameter k > 0 is

upper K u r t left-parenthesis upper X right-parenthesis equals StartFraction minus 6 normal upper Gamma 1 Superscript 4 Baseline plus 12 normal upper Gamma 1 squared normal upper Gamma 2 minus 3 normal upper Gamma 2 squared minus 4 normal upper Gamma 1 normal upper Gamma 3 plus normal upper Gamma 4 Over left-bracket normal upper Gamma 2 minus normal upper Gamma 1 squared right-bracket squared EndFraction

where Γ_i = Γ( 1 + i / k ).

Usage

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

kurtosis( k, lambda )

Returns the excess kurtosis of a Weibull distribution with shape parameter k and scale parameter lambda.

var v = kurtosis( 1.0, 1.0 );
// returns 6.0

v = kurtosis( 4.0, 12.0 );
// returns ~-0.252

v = kurtosis( 8.0, 2.0 );
// returns ~0.328

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 k <= 0, the function returns NaN.

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

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

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

var v = kurtosis( 1.0, 0.0 );
// 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/weibull/kurtosis' );

var lambda;
var k;
var v;
var i;

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