Logarithm of Cumulative Distribution Function

Logistic distribution logarithm of cumulative distribution function.

The cumulative distribution function for a logistic random variable is

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

Usage

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

logcdf( x, mu, s )

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

var y = logcdf( 2.0, 0.0, 1.0 );
// returns ~-0.127

y = logcdf( 0.0, 0.0, 1.0 );
// returns ~-0.693

y = logcdf( -1.0, 4.0, 2.0 );
// returns ~-2.579

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 s < 0, the function returns NaN.

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

If provided s = 0, the function evaluates the logarithm of the CDF of a degenerate distribution centered at mu.

var y = logcdf( 2.0, 8.0, 0.0 );
// returns -Infinity

y = logcdf( 8.0, 8.0, 0.0 );
// returns 0.0

y = logcdf( 10.0, 8.0, 0.0 );
// returns 0.0

logcdf.factory( mu, s )

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

var mylogcdf = logcdf.factory( 10.0, 2.0 );

var y = mylogcdf( 10.0 );
// returns ~-0.693

y = mylogcdf( 8.0 );
// returns ~-1.313

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/logistic/logcdf' );

var mu;
var s;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu() * 10.0;
    mu = randu() * 10.0;
    s = randu() * 10.0;
    y = logcdf( x, mu, s );
    console.log( 'x: %d, µ: %d, s: %d, ln(F(x;µ,s)): %d', x, mu, s, y );
}