Fast Computation of the Non-Central Chi Square PDF Outside the HDR Under a Requisite Precision Constraint

Paul Gendron

doi:10.3390/computation3020326

Abstract

Computation of the non-central chi square probability density function is encountered in diverse fields of applied statistics and engineering. The distribution is commonly computed as a Poisson mixture of central chi square densities, where the terms of the sum are computed starting with the integer nearest the non-centrality parameter. However, for computation of the values in either tail region these terms are not the most significant and starting with them results in an increased computational load without a corresponding increase in accuracy. The most significant terms are shown to be a function of both the non-centrality parameter, the degree of freedom and the point of evaluation. A computationally simple approximate solution to the location of the most significant terms as well as the exact solution based on a Newton–Raphson iteration is presented. A quadratic approximation of the interval of summation is also developed in order to meet a requisite number of significant digits of accuracy. Computationally efficient recursions are used over these improved intervals. The method provides a means of computing the non-central chi square probability density function to a requisite accuracy as a Poisson mixture over all domains of interest.

Highlights

Evaluating the non-central chi square probability density function (PDF) is of practical importance to a number of problems in applied statistics [1]
The three algorithms are compared to illustrate the importance of initialization of the algorithm by proper determination of n∗ (x, ν, λ) in the computation of the non-central chi square PDF in the tail regions
Shown are the Laplace approximate upper (Ub) and lower bounds (Lb) associated with B = 4 used for Algorithm 3

Summary

Introduction

Evaluating the non-central chi square probability density function (PDF) is of practical importance to a number of problems in applied statistics [1]. The non-central chi square density [2], where λ is the non-centrality parameter and ν the degree of freedom, arises in the general case where x = e0 Σ−1 e and e ∼ N (μ, Σ), e ∈ Rν , and Σ positive definite. It follows that x ∼ χ2ν (λ) where λ = μ0 Σ−1 μ. X ∼ χ2ν (λ) denotes that x is distributed as a non-central chi square with ν degrees of freedom and non-centrality parameter λ.

Objectives

Results

Conclusion