Kurtosis

Pareto (Type I) distribution excess kurtosis.

The excess kurtosis for a Pareto (Type I) random variable with shape parameter α and scale parameter β is

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

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

Usage

var kurtosis = require( '@stdlib/math/base/dists/pareto-type1/kurtosis' );

kurtosis( alpha, beta )

Returns the excess kurtosis of a Pareto (Type I) distribution with parameters alpha (shape parameter) and beta (scale parameter).

var v = kurtosis( 5.0, 1.0 );
// returns ~70.8

v = kurtosis( 4.5, 12.0 );
// returns ~146.444

v = kurtosis( 8.0, 2.0 );
// returns ~19.725

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 alpha <= 4, the function returns NaN.

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

v = kurtosis( 3.0, 1.0 );
// returns NaN

v = kurtosis( 4.0, 2.0 );
// returns NaN

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

If provided beta <= 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/pareto-type1/kurtosis' );

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

for ( i = 0; i < 10; i++ ) {
    alpha = ( randu()*10.0 ) + 4.0 + EPS;
    beta = ( randu()*10.0 ) + 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 ) );
}