Logarithm of Cumulative Distribution Function

Evaluate the natural logarithm of the cumulative distribution function (CDF) for a Student's t distribution.

The cumulative distribution function (CDF) for a t distribution random variable is

where v > 0 is the degrees of freedom. 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/t/logcdf' );

logcdf( x, v )

Evaluates the natural logarithm of the cumulative distribution function (CDF) for a Student's t distribution with degrees of freedom v.

var y = logcdf( 2.0, 0.1 );
// returns ~-0.493

y = logcdf( 1.0, 2.0 );
// returns ~-0.237

y = logcdf( -1.0, 4.0 );
// returns ~-1.677

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

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

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

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

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

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

logcdf.factory( v )

Returns a function for evaluating the natural logarithm of the CDF of a Student's t distribution with degrees of freedom v.

var mylogcdf = logcdf.factory( 0.5 );
var y = mylogcdf( 3.0 );
// returns ~-0.203

y = mylogcdf( 1.0 );
// returns ~-0.358

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 logcdf = require( '@stdlib/stats/base/dists/t/logcdf' );

var v;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = (randu() * 6.0) - 3.0;
    v = randu() * 10.0;
    y = logcdf( x, v );
    console.log( 'x: %d, v: %d, ln(F(x;v)): %d', x.toFixed( 4 ), v.toFixed( 4 ), y.toFixed( 4 ) );
}