The Mellin Transform to Manage Quadratic Forms in Normal Random Variables

Abstract

The problem of computing the distribution of quadratic forms in normal variables has a long tradition in the statistical literature. Well-established numerical algorithms that deal with this task rely on the inversion of Fourier transforms or series representations. In this article, the Mellin transform is proposed as a tool to compute both the density and the cumulative distribution functions of a positive definite quadratic form: an outline of the numerical algorithm is presented, providing details on the error analysis. The algorithm’s characteristics allow us to propose an efficient way to compute the random variables’ quantiles. From the theoretical point of view, the analytic properties of the Mellin transform are exploited to provide a novel representation of the distribution of the ratio of independent quadratic forms as a mixture of beta random variables of the second kind. Moreover, algorithms are proposed for computations related to ratios of both independent and dependent quadratic forms. The methods are tested and compared to popular numerical algorithms in terms of computational times and accuracy. The R package QF implementing all the proposed algorithms is also made available. Supplementary materials for this article are available online.