Logarithm of Cumulative Distribution Function

Evaluate the natural logarithm of the cumulative distribution function for a Pareto (Type I) distribution.

The cumulative distribution function for a Pareto (Type I) random variable is

upper F left-parenthesis x right-parenthesis equals 1 minus left-parenthesis StartFraction beta Over x EndFraction right-parenthesis Superscript alpha Baseline for x greater-than-or-equal-to beta

and zero otherwise. In the equation, alpha > 0 is the shape parameter and beta > 0 is the scale parameter.

Usage

var logcdf = require( '@stdlib/math/base/dists/pareto-type1/logcdf' );

logcdf( x, alpha, beta )

Evaluates the natural logarithm of the cumulative distribution function (CDF) for a Pareto (Type I) distribution with parameters alpha (shape parameter) and beta (scale parameter).

var y = logcdf( 2.0, 1.0, 1.0 );
// returns ~-0.693

y = logcdf( 5.0, 2.0, 4.0 );
// returns ~-1.022

y = logcdf( 4.0, 2.0, 2.0 );
// returns ~-0.288

y = logcdf( 1.9, 2.0, 2.0 );
// returns -Infinity

y = logcdf( +Infinity, 4.0, 2.0 );
// returns 0.0

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

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

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

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

If provided alpha <= 0, the function returns NaN.

var y = logcdf( 2.0, -1.0, 0.5 );
// returns NaN

y = logcdf( 2.0, 0.0, 0.5 );
// returns NaN

If provided beta <= 0, the function returns NaN.

var y = logcdf( 2.0, 0.5, -1.0 );
// returns NaN

y = logcdf( 2.0, 0.5, 0.0 );
// returns NaN

logcdf.factory( alpha, beta )

Returns a function for evaluating the natural logarithm of the cumulative distribution function (CDF) of a Pareto (Type I) distribution with parameters alpha (shape parameter) and beta (scale parameter).

var mylogcdf = logcdf.factory( 10.0, 2.0 );
var y = mylogcdf( 3.0 );
// returns ~-0.017

y = mylogcdf( 2.5 );
// returns ~-0.113

Examples

var randu = require( '@stdlib/random/base/randu' );
var logcdf = require( '@stdlib/math/base/dists/pareto-type1/logcdf' );

var alpha;
var beta;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu() * 8.0;
    alpha = randu() * 5.0;
    beta = randu() * 5.0;
    y = logcdf( x, alpha, beta );
    console.log( 'x: %d, α: %d, β: %d, ln(F(x;α,β)): %d', x.toFixed( 4 ), alpha.toFixed( 4 ), beta.toFixed( 4 ), y.toFixed( 4 ) );
}