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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.
*
*
* ## Notice
*
* The original C++ code and copyright notice are from the [Boost library]{@link http://www.boost.org/doc/libs/1_61_0/boost/math/special_functions/beta.hpp}. The implementation has been modified for JavaScript.
*
* ```text
* (C) Copyright John Maddock 2006.
*
* Use, modification and distribution are subject to the
* Boost Software License, Version 1.0. (See accompanying file
* LICENSE or copy at http://www.boost.org/LICENSE_1_0.txt)
* ```
*/
'use strict';
// MODULES //
var binomcoef = require( '@stdlib/math/base/special/binomcoef' );
var floor = require( '@stdlib/math/base/special/floor' );
var pow = require( '@stdlib/math/base/special/pow' );
var MIN_VALUE = require( '@stdlib/constants/float64/smallest-normal' );
// MAIN //
/**
* For integer arguments we can relate the incomplete beta to the complement of the binomial distribution cdf and use this finite sum.
*
* @private
* @param {NonNegativeInteger} n - number of trials
* @param {NonNegativeInteger} k - function input
* @param {Probability} x - function input
* @param {Probability} y - probability equal to `1-x`
* @returns {number} sum
*/
function binomialCCDF( n, k, x, y ) {
var startTerm;
var result;
var start;
var term;
var i;
result = pow( x, n );
if ( result > MIN_VALUE ) {
term = result;
for ( i = floor( n - 1 ); i > k; i-- ) {
term *= ((i + 1) * y) / ((n - i) * x);
result += term;
}
} else {
// First term underflows so we need to start at the mode of the distribution and work outwards:
start = floor( n * x );
if ( start <= k + 1 ) {
start = floor( k + 2 );
}
result = pow( x, start ) * pow( y, n - start );
result *= binomcoef( floor(n), floor(start) );
if ( result === 0.0 ) {
// OK, starting slightly above the mode didn't work, we'll have to sum the terms the old fashioned way:
for ( i = start - 1; i > k; i-- ) {
result += pow( x, i ) * pow( y, n - i );
result *= binomcoef( floor(n), floor(i) );
}
} else {
term = result;
startTerm = result;
for ( i = start - 1; i > k; i-- ) {
term *= ((i + 1) * y) / ((n - i) * x);
result += term;
}
term = startTerm;
for ( i = start + 1; i <= n; i++ ) {
term *= (n - i + 1) * x / (i * y);
result += term;
}
}
}
return result;
}
// EXPORTS //
module.exports = binomialCCDF;
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