Logarithm of Cumulative Distribution Function

Lévy distribution logarithm of cumulative distribution function.

The cumulative distribution function for a Lévy random variable is

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

Usage

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

logcdf( x, mu, c )

Evaluates the logarithm of the cumulative distribution function (CDF) for a Lévy distribution with parameters mu (location parameter) and c > 0 (scale parameter).

var y = logcdf( 2.0, 0.0, 1.0 );
// returns ~-0.735

y = logcdf( 12.0, 10.0, 3.0 );
// returns ~-1.51

y = logcdf( 9.0, 10.0, 3.0 );
// returns -Infinity

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 c <= 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, c )

Returns a function for evaluating the logarithm of the cumulative distribution function of a Lévy distribution with parameters mu (location parameter) and c > 0 (scale parameter).

var mylogcdf = logcdf.factory( 3.0, 1.5 );

var y = mylogcdf( 4.0 );
// returns ~-1.511

y = mylogcdf( 2.0 );
// returns -Infinity

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/levy/logcdf' );

var mu;
var c;
var x;
var y;
var i;

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