Logarithm of Cumulative Distribution Function

Evaluate the natural logarithm of the cumulative distribution function for a Kumaraswamy's double bounded distribution.

The cumulative distribution function for a Kumaraswamy's double bounded random variable is

upper F left-parenthesis x semicolon a comma b right-parenthesis equals 1 minus left-parenthesis 1 minus x Superscript a Baseline right-parenthesis Superscript b

where a > 0 is the first shape parameter and b > 0 is the second shape parameter.

Usage

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

logcdf( x, a, b )

Evaluates the natural logarithm of the cumulative distribution function (CDF) for a Kumaraswamy's double bounded distribution with parameters a (first shape parameter) and b (second shape parameter).

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

y = logcdf( 0.5, 2.0, 4.0 );
// returns ~-0.38

y = logcdf( 0.2, 2.0, 2.0 );
// returns ~-2.546

y = logcdf( 0.8, 4.0, 4.0 );
// returns ~-0.13

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 a <= 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 b <= 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( a, b )

Returns a function for evaluating the natural logarithm of the cumulative distribution function for a Kumaraswamy's double bounded distribution with parameters a (first shape parameter) and b (second shape parameter).

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

var y = mylogcdf( 0.8 );
// returns ~-0.393

y = mylogcdf( 0.3 );
// returns ~-1.116

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/kumaraswamy/logcdf' );

var a;
var b;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu();
    a = ( randu()*5.0 ) + EPS;
    b = ( randu()*5.0 ) + EPS;
    y = logcdf( x, a, b );
    console.log( 'x: %d, a: %d, b: %d, ln(F(x;a,b)): %d', x.toFixed( 4 ), a.toFixed( 4 ), b.toFixed( 4 ), y.toFixed( 4 ) );
}