A branch cut approach to the probability density and distribution functions of a linear combination of central and non-central Chi-square random variables

TL;DR

This paper introduces a novel branch cut method for deriving the probability density function of linear combinations of central and non-central chi-square variables, offering new integral representations and improved numerical accuracy.

Contribution

The paper presents a new branch cut approach for the density function of linear combinations of chi-square variables, enhancing analytical tractability and numerical stability.

Findings

The method provides accurate density function evaluations.

It offers a new integral representation based on branch cuts.

Numerical results demonstrate the approach's effectiveness.

Abstract

The paper considers the distribution of a general linear combination of central and non-central chi-square random variables by exploring the branch cut regions that appear in the standard Laplace inversion process. Due to the original interest from the directional statistics, the focus of this paper is on the density function of such distributions and not on their cumulative distribution function. In fact, our results confirm that the latter is a special case of the former. Our approach provides new insight by generating alternative characterizations of the probability density function in terms of a finite number of feasible univariate integrals. In particular, the central cases seem to allow an interesting representation in terms of the branch cuts, while general degrees of freedom and non-centrality can be easily adopted using recursive differentiation. Numerical results confirm that the proposed approach works well while more transparency and therefore easier control in the accuracy is ensured.