Kurtosis

F distribution excess kurtosis.

The excess kurtosis for a F random variable with numerator degrees of freedom d1 and denominator degrees of freedom d2 is

upper K u r t left-parenthesis upper X right-parenthesis equals gamma 2 equals 12 StartFraction d 1 left-parenthesis 5 d 2 minus 22 right-parenthesis left-parenthesis d 1 plus d 2 minus 2 right-parenthesis plus left-parenthesis d 2 minus 4 right-parenthesis left-parenthesis d 2 minus 2 right-parenthesis squared Over d 1 left-parenthesis d 2 minus 6 right-parenthesis left-parenthesis d 2 minus 8 right-parenthesis left-parenthesis d 1 plus d 2 minus 2 right-parenthesis EndFraction period

for d1 > 0 and d2 > 8. Otherwise, the kurtosis is not defined.

Usage

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

kurtosis( d1, d2 )

Returns the excess kurtosis of a F distribution with parameters d1 (numerator degrees of freedom) and d2 (denominator degrees of freedom).

var v = kurtosis( 4.0, 9.0 );
// returns ~114.273

v = kurtosis( 4.0, 12.0 );
// returns ~26.143

v = kurtosis( 8.0, 9.0 );
// returns ~100.167

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

var v = kurtosis( NaN, 7.0 );
// returns NaN

v = kurtosis( 3.0, NaN );
// returns NaN

If provided d1 <= 0, the function returns NaN.

var v = kurtosis( 1.0, 9.0 );
// returns NaN

v = kurtosis( -1.0, 9.0 );
// returns NaN

If provided d2 <= 8, the function returns NaN.

var v = kurtosis( 1.0, 8.0 );
// returns NaN

v = kurtosis( 2.2, 5.5 );
// returns NaN

v = kurtosis( 3.0, -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/f/kurtosis' );

var d1;
var d2;
var v;
var i;

for ( i = 0; i < 10; i++ ) {
    d1 = ( randu()*10.0 ) + EPS;
    d2 = ( randu()*20.0 ) + EPS;
    v = kurtosis( d1, d2 );
    console.log( 'd1: %d, d2: %d, skew(X;d1,d2): %d', d1.toFixed( 4 ), d2.toFixed( 4 ), v.toFixed( 4 ) );
}