Quantile Function

Cauchy distribution quantile function.

The quantile function for a Cauchy random variable is

upper Q left-parenthesis p semicolon x 0 comma gamma right-parenthesis equals x 0 plus gamma tangent left-bracket pi left-parenthesis p minus one-half right-parenthesis right-bracket

for 0 <= p <= 1, where x0 is the location parameter and gamma > 0 is the scale parameter.

Usage

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

quantile( p, x0, gamma )

Evaluates the quantile function for a Cauchy distribution with parameters x0 (location parameter) and gamma > 0 (scale parameter).

var y = quantile( 0.5, 0.0, 1.0 );
// returns 0.0

y = quantile( 0.2, 4.0, 2.0 );
// returns ~1.247

y = quantile( 0.9, 4.0, 2.0 );
// returns ~10.155

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

var y = quantile( 1.9, 0.0, 1.0 );
// returns NaN

y = quantile( -0.1, 0.0, 1.0 );
// returns NaN

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

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

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

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

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

var y = quantile( 0.4, 0.0, -1.0 );
// returns NaN

y = quantile( 0.4, 0.0, 0.0 );
// returns NaN

quantile.factory( x0, gamma )

Returns a function for evaluating the quantile function of a Cauchy distribution with location parameter x0 and scale parameter gamma > 0.

var myquantile = quantile.factory( 10.0, 2.0 );

var y = myquantile( 0.2 );
// returns ~8.317

y = myquantile( 0.8 );
// returns ~11.683

Examples

var randu = require( '@stdlib/random/base/randu' );
var EPS = require( '@stdlib/constants/math/float64-eps' );
var quantile = require( '@stdlib/math/base/dists/cauchy/quantile' );

var gamma;
var x0;
var p;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    p = randu();
    x0 = ( randu()*10.0 ) - 5.0;
    gamma = ( randu()*20.0 ) + EPS;
    y = quantile( p, gamma, x0 );
    console.log( 'p: %d, x0: %d, γ: %d, Q(p;x0,γ): %d', p.toFixed(4), x0.toFixed(4), gamma.toFixed(4), y.toFixed(4) );
}