Quantile Function

Gamma distribution quantile function.

The quantile function for a gamma random variable is

upper Q left-parenthesis p semicolon alpha comma beta right-parenthesis equals StartFraction 1 Over beta EndFraction upper P Superscript negative 1 Baseline left-parenthesis p comma alpha right-parenthesis

for 0 <= p < 1, where alpha is the shape parameter and beta is the rate parameter of the distribution. P^{-1} is the inverse of the lower regularized incomplete gamma function.

Usage

var quantile = require( '@stdlib/math/base/dists/gamma/quantile' );

quantile( p, alpha, beta )

Evaluates the quantile function for a gamma distribution with parameters alpha (shape parameter) and beta (rate parameter).

var y = quantile( 0.8, 2.0, 1.0 );
// returns ~2.994

y = quantile( 0.5, 4.0, 2.0 );
// returns ~1.836

If provided a probability p outside the interval [0,1], the function returns NaN.

var y = quantile( 1.9, 1.0, 1.0 );
// returns NaN

y = quantile( -0.1, 1.0, 1.0 );
// returns NaN

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

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

y = quantile( 0.0, NaN, 1.0 );
// returns NaN

y = quantile( 0.0, 1.0, NaN );
// returns NaN

If provided alpha < 0, the function returns NaN.

var y = quantile( 0.4, -1.0, 1.0 );
// returns NaN

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

var y = quantile( 0.3, 0.0, 2.0 );
// returns 0.0

y = quantile( 0.9, 0.0, 2.0 );
// returns 0.0

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

var y = quantile( 0.4, 1.0, -1.0 );
// returns NaN

quantile.factory( alpha, beta )

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

var myquantile = quantile.factory( 2.0, 2.0 );
var y = myquantile( 0.8 );
// returns ~1.497

y = myquantile( 0.4 );
// returns ~0.688

Examples

var randu = require( '@stdlib/random/base/randu' );
var quantile = require( '@stdlib/math/base/dists/gamma/quantile' );

var alpha;
var beta;
var p;
var y;
var i;

for ( i = 0; i < 20; i++ ) {
    p = randu();
    alpha = randu() * 5.0;
    beta = randu() * 5.0;
    y = quantile( p, alpha, beta );
    console.log( 'p: %d, α: %d, β: %d, Q(p;α,β): %d', p.toFixed( 4 ), alpha.toFixed( 4 ), beta.toFixed( 4 ), y.toFixed( 4 ) );
}