Non-Central Multivariate Chi-Square and Gamma Distributions

TL;DR

This paper derives integral representations for the distribution functions of multivariate non-central gamma and chi-square distributions, including special cases with specific correlation structures, facilitating their computation.

Contribution

It introduces new integral formulas for multivariate non-central gamma and chi-square distributions, especially for low dimensions and specific correlation matrices, expanding computational methods.

Findings

Derived (p-1)-variate integral representations for distribution functions.

Provided detailed formulas for p=2 and p=3 cases.

Presented alternative formulas for one-factorial correlation matrices.

Abstract

A (p-1)-variate integral representation is given for the cumulative distribution function of the general p-variate non-central gamma distribution with a non-centrality matrix of any admissible rank. The real part of products of well known analytical functions is integrated over arguments from (-pi,pi). To facilitate the computation, these formulas are given more detailed for p=2 and p=3. These (p-1)-variate integrals are also derived for the diagonal of a non-central complex Wishart Matrix. Furthermore, some alternative formulas are given for the cases with an associated "one-factorial" (pxp)-correlation matrix R, i.e. R differs from a suitable diagonal matrix only by a matrix of rank 1, which holds in particular for all (3x3)-R with no vanishing correlation.