Logarithm of Probability Density Function

Fréchet distribution logarithm of probability density function.

The probability density function for a Fréchet random variable is

f left-parenthesis x semicolon mu comma beta right-parenthesis equals StartFraction alpha Over s EndFraction left-parenthesis StartFraction x minus m Over s EndFraction right-parenthesis Superscript negative 1 minus alpha Baseline e Superscript minus left-parenthesis StartFraction x minus m Over s EndFraction right-parenthesis Super Superscript negative alpha

where α > 0 is the shape, s > 0 the scale and m the location parameter.

Usage

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

logpdf( x, alpha, s, m )

Evaluates the logarithm of the probability density function (PDF) for a Fréchet distribution with shape alpha, scale s, and location m at a value x.

var y = logpdf( 10.0, 2.0, 3.0, 5.0 );
// returns ~-2.298

y = logpdf( -3.0, 1.0, 2.0, -4.0 );
// returns ~-1.307

y = logpdf( 0.0, 2.0, 1.0, -1.0 );
// returns ~-0.307

If provided x <= m, the function returns -Infinity.

y = logpdf( -2.0, 2.0, 1.0, -1.0 );
// returns -Infinity

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

var y = logpdf( NaN, 1.0, 1.0, 0.0 );
// returns NaN

y = logpdf( 0.0, NaN, 1.0, 0.0 );
// returns NaN

y = logpdf( 0.0, 1.0, NaN, 0.0);
// returns NaN

y = logpdf( 0.0, 1.0, 1.0, NaN );
// returns NaN

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

var y = logpdf( 2.0, -0.1, 1.0, 1.0 );
// returns NaN

y = logpdf( 2.0, 0.0, 1.0, 1.0 );
// returns NaN

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

var y = logpdf( 2.0, 1.0, -1.0, 1.0 );
// returns NaN

y = logpdf( 2.0, 1.0, 0.0, 1.0 );
// returns NaN

logpdf.factory( alpha, s, m )

Returns a function for evaluating the logarithm of the probability density function of a Fréchet distribution with shape alpha, scale s, and location m.

var mylogpdf = logpdf.factory( 3.0, 3.0, 5.0 );

var y = mylogpdf( 10.0 );
// returns ~-2.259

y = mylogpdf( 7.0 );
// returns ~-1.753

Examples

var randu = require( '@stdlib/random/base/randu' );
var logpdf = require( '@stdlib/math/base/dists/frechet/logpdf' );

var alpha;
var m;
var s;
var x;
var y;
var i;

for ( i = 0; i < 100; i++ ) {
    alpha = randu() * 10.0;
    x = randu() * 10.0;
    s = randu() * 10.0;
    m = randu() * 10.0;
    y = logpdf( x, alpha, s, m );
    console.log( 'x: %d, α: %d, s: %d, m: %d, ln(f(x;α,s,m)): %d', x.toFixed( 4 ), alpha.toFixed( 4 ), s.toFixed( 4 ), m.toFixed( 4 ), y.toFixed( 4 ) );
}