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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 betainc = require( '@stdlib/math/base/special/betainc' );
var isnan = require( '@stdlib/math/base/assert/is-nan' );
var PINF = require( '@stdlib/constants/float64/pinf' );
// MAIN //
/**
* Evaluates the cumulative distribution function (CDF) for an F distribution with numerator degrees of freedom `d1` and denominator degrees of freedom `d2` at a value `x`.
*
* ## Method
*
* The CDF is the regularized incomplete beta function evaluated at \\(z = d_1 x / (d_2 + d_1 x)\\):
*
* ```tex
* F(x;d_1,d_2) = I_{z}\left(\tfrac{d_1}{2}, \tfrac{d_2}{2}\right) = 1 - I_{1-z}\left(\tfrac{d_2}{2}, \tfrac{d_1}{2}\right)
* ```
*
* To avoid rounding an argument close to one, the function passes \\(z\\) to the lower tail when \\(z \leq 1/2\\) (i.e., \\(d_1 x \leq d_2\\)), and \\(1-z = d_2/(d_2 + d_1 x)\\) to the upper tail otherwise.
*
* @param {number} x - input value
* @param {PositiveNumber} d1 - numerator degrees of freedom
* @param {PositiveNumber} d2 - denominator degrees of freedom
* @returns {Probability} evaluated CDF
*
* @example
* var y = cdf( 2.0, 1.0, 1.0 );
* // returns ~0.608
*
* @example
* var y = cdf( 2.0, 8.0, 4.0 );
* // returns ~0.737
*
* @example
* var y = cdf( -1.0, 2.0, 2.0 );
* // returns 0.0
*
* @example
* var y = cdf( +Infinity, 4.0, 2.0 );
* // returns 1.0
*
* @example
* var y = cdf( -Infinity, 4.0, 2.0 );
* // returns 0.0
*
* @example
* var y = cdf( NaN, 1.0, 1.0 );
* // returns NaN
*
* @example
* var y = cdf( 0.0, NaN, 1.0 );
* // returns NaN
*
* @example
* var y = cdf( 0.0, 1.0, NaN );
* // returns NaN
*
* @example
* var y = cdf( 2.0, 1.0, -1.0 );
* // returns NaN
*
* @example
* var y = cdf( 2.0, -1.0, 1.0 );
* // returns NaN
*/
function cdf( x, d1, d2 ) {
if (
isnan( x ) ||
isnan( d1 ) ||
isnan( d2 ) ||
d1 <= 0.0 ||
d2 <= 0.0
) {
return NaN;
}
if ( x <= 0.0 ) {
return 0.0;
}
if ( x === PINF ) {
return 1.0;
}
if ( d1 * x > d2 ) {
return betainc( d2/(d2+(d1*x)), d2/2.0, d1/2.0, true, true );
}
return betainc( (d1*x)/(d2+(d1*x)), d1/2.0, d2/2.0, true, false );
}
// EXPORTS //
module.exports = cdf;
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