Logarithm of Cumulative Distribution Function

Cauchy distribution logarithm of cumulative distribution function.

The cumulative distribution function for a Cauchy random variable is

where x0 is the location parameter and gamma > 0 is the scale parameter.

Usage

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

logcdf( x, x0, gamma )

Evaluates the natural logarithm of the cumulative distribution function (CDF) for a Cauchy distribution with parameters x0 (location parameter) and gamma > 0 (scale parameter).

var y = logcdf( 4.0, 0.0, 2.0 );
// returns ~-0.16

y = logcdf( 1.0, 0.0, 2.0 );
// returns ~-0.435

y = logcdf( 1.0, 3.0, 2.0 );
// returns ~-1.386

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

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

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

y = logcdf( 1.0, NaN, 3.0 );
// returns NaN

If provided gamma <= 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( x0, gamma )

Returns a function for evaluating the natural logarithm of the cumulative distribution function of a Cauchy distribution with parameters x0 (location parameter) and gamma > 0 (scale parameter).

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

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

y = mylogcdf( 12.0 );
// returns ~-0.288

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 EPS = require( '@stdlib/constants/float64/eps' );
var logcdf = require( '@stdlib/stats/base/dists/cauchy/logcdf' );

var gamma;
var x0;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu() * 10.0;
    x0 = randu() * 10.0;
    gamma = ( randu()*10.0 ) + EPS;
    y = logcdf( x, x0, gamma );
    console.log( 'x: %d, x0: %d, γ: %d, ln(F(x;x0,γ)): %d', x, x0, gamma, y );
}