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* @license Apache-2.0
*
* Copyright (c) 2026 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 constantFunction = require( '@stdlib/utils/constant-function' );
var degenerate = require( '@stdlib/stats/base/dists/degenerate/cdf' ).factory;
var erfcx = require( '@stdlib/math/base/special/erfcx' );
var isnan = require( '@stdlib/math/base/assert/is-nan' );
var sqrt = require( '@stdlib/math/base/special/sqrt' );
var erfc = require( '@stdlib/math/base/special/erfc' );
var exp = require( '@stdlib/math/base/special/exp' );
var PINF = require( '@stdlib/constants/float64/pinf' );
// MAIN //
/**
* Returns a function for evaluating the cumulative distribution function (CDF) for a Wald distribution.
*
* @param {PositiveNumber} mu - mean
* @param {NonNegativeNumber} lambda - shape parameter
* @returns {Function} function to evaluate the cumulative distribution function
*
* @example
* var cdf = factory( 1.0, 1.0 );
* var y = cdf( 2.0 );
* // returns ~0.885
*
* y = cdf( 0.5 );
* // returns ~0.365
*/
function factory( mu, lambda ) {
if (
isnan( mu ) ||
isnan( lambda ) ||
mu <= 0.0 ||
lambda < 0.0
) {
return constantFunction( NaN );
}
if ( lambda === 0.0 ) {
return degenerate( mu );
}
return cdf;
/**
* Evaluates the cumulative distribution function (CDF) of a Wald distribution.
*
* @private
* @param {number} x - input value
* @returns {Probability} evaluated CDF
*
* @example
* var y = cdf( 2.0 );
* // returns <number>
*/
function cdf( x ) {
var t1;
var t2;
var a;
var b;
var z;
if ( isnan( x ) ) {
return NaN;
}
if ( x <= 0.0 ) {
return 0.0;
}
if ( x === PINF ) {
return 1.0;
}
z = sqrt( lambda / x );
a = ( z * ( ( x / mu ) - 1.0 ) ) / sqrt( 2.0 );
b = ( z * ( ( x / mu ) + 1.0 ) ) / sqrt( 2.0 );
// Φ(a) = 0.5 * erfc( -a / √2 )
t1 = 0.5 * erfc( -a );
// exp( 2λ/μ ) * erfc( b ) = erfcx( b ) * exp( -a² ), as b² - a² = 2λ/μ; computing via `erfcx` avoids overflow of the exponential term for large `λ/μ`:
t2 = 0.5 * erfcx( b ) * exp( -a * a );
return t1 + t2;
}
}
// EXPORTS //
module.exports = factory;
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