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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 isInteger = require( '@stdlib/math/base/assert/is-integer' );
var isnan = require( '@stdlib/math/base/assert/is-nan' );
var ceil = require( '@stdlib/math/base/special/ceil' );
// MAIN //
/**
* Evaluates the quantile function for a discrete uniform distribution with minimum support `a` and maximum support `b` at a probability `p`.
*
* ## Method
*
* The quantile is the smallest \\(x\\) in the support with \\(F(x) \geq p\\). As \\(F(x) = (x-a+1)/n\\), where \\(n = b-a+1\\), this is
*
* ```tex
* Q(p) = a - 1 + \lceil p n \rceil
* ```
*
* for \\(p > 0\\), and \\(Q(0) = a\\). Because \\(pn\\) can round up past an integer (e.g., \\(0.3 \cdot 10 = 3.0000000000000004\\)), the function steps \\(k = \lceil pn \rceil\\) back by one when \\((k-1)/n \geq p\\), which is the same comparison the CDF makes.
*
* @param {Probability} p - input value
* @param {integer} a - minimum support
* @param {integer} b - maximum support
* @returns {integer} evaluated quantile function
*
* @example
* var y = quantile( 0.8, 0, 1 );
* // returns 1
*
* @example
* var y = quantile( 0.5, 0, 10 );
* // returns 5
*
* @example
* var y = quantile( 1.1, 0, 1 );
* // returns NaN
*
* @example
* var y = quantile( -0.2, 0, 1 );
* // returns NaN
*
* @example
* var y = quantile( 0.5, 1, 1.5 );
* // returns NaN
*
* @example
* var y = quantile( NaN, 0, 1 );
* // returns NaN
*
* @example
* var y = quantile( 0.0, NaN, 1.0 );
* // returns NaN
*
* @example
* var y = quantile( 0.0, 0, NaN );
* // returns NaN
*
* @example
* var y = quantile( 0.5, 2, 1 );
* // returns NaN
*/
function quantile( p, a, b ) {
var n;
var k;
if (
isnan( p ) ||
isnan( a ) ||
isnan( b ) ||
!isInteger( a ) ||
!isInteger( b ) ||
a > b ||
p < 0.0 ||
p > 1.0
) {
return NaN;
}
if ( p === 0.0 ) {
return a;
}
n = b - a + 1;
k = ceil( p * n );
if ( ( k-1 ) / n >= p ) {
k -= 1;
}
return a + k - 1;
}
// EXPORTS //
module.exports = quantile;
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