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* @license Apache-2.0
*
* Copyright (c) 2018 The Stdlib Authors.
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
'use strict';
// MODULES //
var isNonNegativeInteger = require( '@stdlib/math/base/assert/is-nonnegative-integer' );
var erfcinv = require( '@stdlib/math/base/special/erfcinv' );
var isnan = require( '@stdlib/math/base/assert/is-nan' );
var round = require( '@stdlib/math/base/special/round' );
var sqrt = require( '@stdlib/math/base/special/sqrt' );
var cdf = require( '@stdlib/stats/base/dists/binomial/cdf' );
var SQRT2 = require( '@stdlib/constants/float64/sqrt-two' );
var PINF = require( '@stdlib/constants/float64/pinf' );
var searchLeft = require( './search_left.js' );
var searchRight = require( './search_right.js' );
// MAIN //
/**
* Evaluates the quantile function for a binomial distribution with number of trials `n` and success probability `p` at a probability `r`.
*
* @param {Probability} r - input value
* @param {NonNegativeInteger} n - number of trials
* @param {Probability} p - success probability
* @returns {NonNegativeInteger} evaluated quantile function
*
* @example
* var y = quantile( 0.4, 20, 0.2 );
* // returns 3
*
* @example
* var y = quantile( 0.8, 20, 0.2 );
* // returns 5
*
* @example
* var y = quantile( 0.5, 10, 0.4 );
* // returns 4
*
* @example
* var y = quantile( 0.0, 10, 0.4 );
* // returns 0
*
* @example
* var y = quantile( 1.0, 10, 0.4 );
* // returns 10
*
* @example
* var y = quantile( NaN, 20, 0.5 );
* // returns NaN
*
* @example
* var y = quantile( 0.2, NaN, 0.5 );
* // returns NaN
*
* @example
* var y = quantile( 0.2, 20, NaN );
* // returns NaN
*
* @example
* var y = quantile( 0.5, 1.5, 0.5 );
* // returns NaN
*
* @example
* var y = quantile( 0.5, -2.0, 0.5 );
* // returns NaN
*
* @example
* var y = quantile( 0.5, 20, -1.0 );
* // returns NaN
*
* @example
* var y = quantile( 0.5, 20, 1.5 );
* // returns NaN
*/
function quantile( r, n, p ) {
var sigmaInv;
var guess;
var sigma;
var corr;
var mu;
var x2;
var x;
if (
isnan( r ) ||
isnan( n ) ||
isnan( p ) ||
r < 0.0 ||
r > 1.0 ||
p < 0.0 ||
p > 1.0 ||
!isNonNegativeInteger( n ) ||
n === PINF
) {
return NaN;
}
if ( r === 1.0 || p === 1.0 ) {
return n;
}
if ( r === 0.0 || p === 0.0 || n === 0 ) {
return 0.0;
}
// Cornish-Fisher expansion:
mu = n * p;
sigma = sqrt( n * p * ( 1.0-p ) );
sigmaInv = 1.0 / sigma;
if ( r < 0.5 ) {
x = -erfcinv( 2.0 * r ) * SQRT2;
} else {
x = erfcinv( 2.0 * ( 1.0-r ) ) * SQRT2;
}
x2 = x * x;
// Skewness correction:
corr = x + ( sigmaInv * ( x2-1.0 ) / 6.0 );
guess = round( mu + (sigma * corr) );
if ( cdf( guess, n, p ) >= r ) {
return searchLeft( guess, r, n, p );
}
return searchRight( guess, r, n, p );
}
// EXPORTS //
module.exports = quantile;
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