Logarithm of Cumulative Distribution Function

Beta distribution logarithm of cumulative distribution function.

The cumulative distribution function for a beta random variable is

where alpha > 0 is the first shape parameter and beta > 0 is the second shape parameter. In the definition, Beta( x; a, b ) denotes the lower incomplete beta function and Beta( a, b ) the beta function.

Usage

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

logcdf( x, alpha, beta )

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

var y = logcdf( 0.5, 1.0, 1.0 );
// returns ~-0.693

y = logcdf( 0.5, 2.0, 4.0 );
// returns ~-0.208

y = logcdf( 0.2, 2.0, 2.0 );
// returns ~-2.263

y = logcdf( 0.8, 4.0, 4.0 );
// returns ~-0.034

y = logcdf( -0.5, 4.0, 2.0 );
// returns -Infinity

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

y = logcdf( 1.5, 4.0, 2.0 );
// returns 0.0

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 for a beta distribution with parameters alpha (first shape parameter) and beta (second shape parameter).

var mylogcdf = logcdf.factory( 0.5, 0.5 );

var y = mylogcdf( 0.8 );
// returns ~-0.35

y = mylogcdf( 0.3 );
// returns ~-0.997

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/beta/logcdf' );

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

for ( i = 0; i < 10; i++ ) {
    x = randu();
    alpha = ( randu()*5.0 ) + EPS;
    beta = ( randu()*5.0 ) + EPS;
    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 ) );
}