Kurtosis

Fréchet distribution kurtosis.

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

upper K u r t equals StartLayout Enlarged left-brace 1st Row 1st Column negative 6 plus StartStartFraction normal upper Gamma left-parenthesis 1 minus StartFraction 4 Over alpha EndFraction right-parenthesis minus 4 normal upper Gamma left-parenthesis 1 minus StartFraction 3 Over alpha EndFraction right-parenthesis normal upper Gamma left-parenthesis 1 minus StartFraction 1 Over alpha EndFraction right-parenthesis plus 3 normal upper Gamma squared left-parenthesis 1 minus StartFraction 2 Over alpha EndFraction right-parenthesis OverOver left-bracket normal upper Gamma left-parenthesis 1 minus StartFraction 2 Over alpha EndFraction right-parenthesis minus normal upper Gamma squared left-parenthesis 1 minus StartFraction 1 Over alpha EndFraction right-parenthesis right-bracket squared EndEndFraction 2nd Column for alpha greater-than 4 2nd Row 1st Column normal infinity 2nd Column otherwise EndLayout

where Γ is the gamma function.

Usage

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

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

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

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 ) );
}