An Accurate Approximation to the Distribution of a Linear Combination of Non-Central Chi-Square Random Variables

Abstract

This paper provides an accessible methodology for approximating the distribution of a general linear combination of non-central chi-square random variables. Attention is focused on the main application of the results, namely the distribution of positive definite and indefinite quadratic forms in normal random variables. After explaining that the moments of a quadratic form can be determined from its cumulants by means of a recursive formula, we propose a moment-based approximation of the density function of a positive definite quadratic form, which consists of a gamma density function that is adjusted by a linear combination of Laguerre polynomials or, equivalently, by a single polynomial. On expressing an indefinite quadratic form as the difference of two positive definite quadratic forms, explicit representations of approximations to its density and distribution functions are obtained in terms of confluent hypergeometric functions. The proposed closed form expressions converge rapidly and provide accurate approximations over the entire support of the distribution. Additionally, bounds are derived for the integrated squared and absolute truncation errors. An easily implementable algorithm is provided and several illustrative numerical examples are presented. In particular, the methodology is applied to the Durbin–Watson statistic. Finally, relevant computational considerations are discussed. Linear combinations of chi-square random variables and quadratic forms in normal variables being ubiquitous in statistics, the distribution approximation technique introduced herewith should prove widely applicable.