Logarithm of Cumulative Distribution Function

Gamma distribution logarithm of cumulative distribution function (CDF).

The cumulative distribution function for a gamma random variable is

where alpha is the shape parameter and beta is the rate parameter of the distribution. gamma is the lower incomplete gamma function.

Usage

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

logcdf( x, alpha, beta )

Evaluates the natural logarithm of the cumulative distribution function (CDF) for a gamma distribution with parameters alpha (shape parameter) and beta (rate parameter).

var y = logcdf( 2.0, 0.5, 1.0 );
// returns ~-0.047

y = logcdf( 0.1, 1.0, 1.0 );
// returns ~-2.352

y = logcdf( -1.0, 4.0, 2.0 );
// returns -Infinity

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, -0.5, 1.0 );
// returns NaN

If provided alpha = 0, the function evaluates the logarithm of the CDF for a degenerate distribution centered at 0.

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

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

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

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

var y = logcdf( 2.0, 1.0, 0.0 );
// returns NaN

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

logcdf.factory( alpha, beta )

Returns a function for evaluating the natural logarithm of the CDF for a gamma distribution with parameters alpha (shape parameter) and beta (rate parameter).

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

var y = mylogcdf( 1.0 );
// returns ~-1.655

y = mylogcdf( 4.0 );
// returns ~-0.064

Examples

var randu = require( '@stdlib/random/base/randu' );
var logcdf = require( '@stdlib/stats/base/dists/gamma/logcdf' );

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

for ( i = 0; i < 10; i++ ) {
    x = randu() * 3.0;
    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 ) );
}