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.

Installation

npm install @stdlib/stats-base-dists-cauchy-logcdf

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 );
}

Notice

This package is part of stdlib, a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.

Community

License

See LICENSE.

Copyright

Copyright © 2016-2024. The Stdlib Authors.