Logarithm of Cumulative Distribution Function

Gumbel distribution logarithm of cumulative distribution function.

The cumulative distribution function for a Gumbel random variable is

where mu is the location parameter and beta > 0 is the scale parameter.

Usage

var logcdf = require( '@stdlib/stats/base/dists/gumbel/logcdf' );

logcdf( x, mu, beta )

Evaluates the logarithm of the cumulative distribution function (CDF) for a Gumbel distribution with parameters mu (location parameter) and beta (scale parameter).

var y = logcdf( 10.0, 0.0, 3.0 );
// returns ~-0.036

y = logcdf( -2.0, 0.0, 3.0 );
// returns ~-1.948

y = logcdf( 0.0, 0.0, 1.0 );
// returns ~-1

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

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

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

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

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

var y = logcdf( 2.0, 0.0, -1.0 );
// returns NaN

y = logcdf( 2.0, 0.0, 0.0 );
// returns NaN

logcdf.factory( mu, beta )

Returns a function for evaluating the logarithm of the cumulative distribution function of a Gumbel distribution with parameters mu (location parameter) and beta (scale parameter).

var mylogcdf = logcdf.factory( 0.0, 3.0 );

var y = mylogcdf( 10.0 );
// returns ~-0.036

y = mylogcdf( -2.0 );
// returns ~-1.948

Notes

In virtually all cases, using the logpdf or logcdf functions is preferable to manually computing the logarithm of the pdf or cdf, respectively, since the latter is prone to overflow and underflow.

Examples

var randu = require( '@stdlib/random/base/randu' );
var logcdf = require( '@stdlib/stats/base/dists/gumbel/logcdf' );

var beta;
var mu;
var x;
var y;
var i;

for ( i = 0; i < 100; i++ ) {
    x = randu() * 10.0;
    mu = randu() * 10.0;
    beta = randu() * 10.0;
    y = logcdf( x, mu, beta );
    console.log( 'x: %d, µ: %d, β: %d, ln(F(x;µ,β)): %d', x.toFixed( 4 ), mu.toFixed( 4 ), beta.toFixed( 4 ), y.toFixed( 4 ) );
}