Kurtosis

Lognormal distribution excess kurtosis.

The excess kurtosis for a lognormal random variable with location parameter μ and scale parameter σ > 0 is

upper K u r t left-parenthesis upper X right-parenthesis equals exp left-parenthesis 4 sigma squared right-parenthesis plus 2 exp left-parenthesis 3 sigma squared right-parenthesis plus 3 exp left-parenthesis 2 sigma squared right-parenthesis minus 6

According to the definition, the natural logarithm of a random variable from a lognormal distribution follows a normal distribution.

Usage

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

kurtosis( mu, sigma )

Returns the excess kurtosis for a lognormal distribution with location mu and scale sigma.

var y = kurtosis( 2.0, 1.0 );
// returns ~110.936

y = kurtosis( 0.0, 1.0 );
// returns ~110.936

y = kurtosis( -1.0, 4.0 );
// returns 6.235150484159035e+27

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

var y = kurtosis( NaN, 1.0 );
// returns NaN

y = kurtosis( 0.0, NaN );
// returns NaN

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

var y = kurtosis( 0.0, 0.0 );
// returns NaN

y = kurtosis( 0.0, -1.0 );
// returns NaN

Examples

var randu = require( '@stdlib/random/base/randu' );
var kurtosis = require( '@stdlib/math/base/dists/lognormal/kurtosis' );

var sigma;
var mu;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    mu = ( randu()*10.0 ) - 5.0;
    sigma = randu() * 20.0;
    y = kurtosis( mu, sigma );
    console.log( 'µ: %d, σ: %d, Kurt(X;µ,σ): %d', mu.toFixed( 4 ), sigma.toFixed( 4 ), y.toFixed( 4 ) );
}