Logarithm of Cumulative Distribution Function

Lévy distribution logarithm of cumulative distribution function.

The cumulative distribution function for a Lévy random variable is

upper F left-parenthesis x semicolon mu comma b right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column e r f c left-parenthesis StartRoot StartFraction c Over 2 left-parenthesis x minus mu right-parenthesis EndFraction EndRoot right-parenthesis 2nd Column for x greater-than mu 2nd Row 1st Column 0 2nd Column otherwise EndLayout

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

Usage

var logcdf = require( '@stdlib/math/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 ~-0.734

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 ~-1.514

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

Examples

var randu = require( '@stdlib/random/base/randu' );
var EPS = require( '@stdlib/constants/math/float64-eps' );
var logcdf = require( '@stdlib/math/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 ) );
}