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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

$$F(x;\nu) = 1 - \frac{1}{2} \frac{\mathop{\mathrm{Beta}}(\tfrac{\nu}{\nu + x^2};\,\tfrac{\nu}{2},\tfrac{1}{2})}{\mathop{\mathrm{Beta}}(\tfrac{\nu}{2}, \tfrac{1}{2})}$$

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.

Installation

npm install @stdlib/stats-base-dists-t-logcdf

Alternatively,

• To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
• If you are using Deno, visit the deno branch (see README for usage intructions).
• For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

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 ) );
}

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Notice

This package is part of stdlib, a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.

Community

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License

See LICENSE.

Copyright

Copyright © 2016-2024. The Stdlib Authors.