[Paper Review] Integral representations for convolutions of non-central multivariate gamma distributions

Thomas Royen|ArXiv.org|Apr 4, 2007

Statistical Distribution Estimation and Applications12 references3 citations

TL;DR

This paper derives three novel integral representations for the cumulative distribution functions (CDFs) of convolutions of non-central multivariate gamma distributions using complex integration over the p-cube $(-\pi, \pi]^p$. The key contribution is numerically efficient CDF computation via $p$-dimensional integrals, particularly for the joint distribution of diagonal elements of generalized quadratic forms in normal random matrices, offering computational advantages over series expansions or Fourier/Laplace inversion.

ABSTRACT

Three types of integral representations for the cumulative distribution functions of convolutions of non-central p-variate gamma distributions are given by integration of elementary complex functions over the p-cube Cp = (-pi,pi]x...x(-pi,pi]. In particular, the joint distribution of the diagonal elements of a generalized quadratic form XAX' with n independent normally distributed column vectors in X is obtained. For a single p-variate gamma distribution function (p-1)-variate integrals over Cp-1 are derived. The integrals are numerically more favourable than integrals obtained from the Fourier or laplace inversion formula.

Motivation & Objective

• To develop numerically efficient representations for the CDFs of convolutions of non-central multivariate gamma distributions.
• To address the computational infeasibility of series expansions for non-central multivariate gamma distributions.
• To provide a practical alternative to Fourier or Laplace inversion for computing CDFs of generalized quadratic forms in normal random matrices.
• To derive explicit integral formulas over the $p$-cube $(-\pi, \pi]^p$ that are more favorable for numerical evaluation than existing methods.

Proposed method

• Utilizes complex analysis techniques, specifically contour integration over the $p$-cube $(-\pi, \pi]^p$, to transform Laplace transforms into CDF representations.
• Applies a general method from theorem 1 to convert products of generating functions into integrals via $ (2\pi)^{-p} \int_{\mathcal{C}_p} A(y) B(y^{-1}) \, d\varphi_1 \cdots d\varphi_p $.
• Employs analytic continuation and residue calculus to derive integral forms for the CDFs of $\Gamma_p(\alpha_k, \Sigma_k, \Delta_k)$-distributed random vectors.
• Derives $p$-dimensional integrals for the CDF of the diagonal elements of $XAX^\top$ with $X$ having i.i.d. $N_p(\mu_k, \Sigma)$-distributed columns.
• Reduces the dimensionality of the integral to $p-1$ variables for the single $\Gamma_p(\alpha, \Sigma, \Delta)$ case via recursive decomposition.
• Uses lemma 2 and lemma 3 to handle matrix determinants and traces in the exponent, enabling transformation of complex integrals into tractable forms.

Experimental results

Research questions

• RQ1 Can the CDF of a convolution of non-central multivariate gamma distributions be represented via a complex integral over the $p$-cube $(-\pi, \pi]^p$?
• RQ2 How can the joint distribution of the diagonal elements of a generalized quadratic form $XAX^\top$ be expressed in terms of multivariate gamma CDFs?
• RQ3 Are these integral representations numerically more favorable than series expansions or Fourier/Laplace inversion for non-central multivariate gamma distributions?
• RQ4 What is the structure of the integrand and under what conditions does the integral converge for non-integer $\alpha$?

Key findings

• Three distinct types of integral representations for the CDF of convolutions of non-central multivariate gamma distributions are derived, all integrating over $(-\pi, \pi]^p$.
• For a single $\Gamma_p(\alpha, \Sigma, \Delta)$ distribution, a $(p-1)$-dimensional integral over $(-\pi, \pi]^{p-1}$ is derived, reducing computational complexity.
• The integrals are shown to be numerically more favorable than series expansions, especially when the latter involve intricate coefficients or slow convergence.
• The joint distribution of the diagonal elements of $XAX^\top$, with $X$ having i.i.d. normal columns and $A \geq 0$, is expressed as a sum of $q$ independent $\Gamma_p(\frac{1}{2}, \lambda_k\Sigma, \Delta_k)$-distributed vectors.
• The method enables exact CDF computation without relying on infinite series or numerical inversion of characteristic functions.
• The approach is validated through transformation of generating functions and residue calculus, with explicit verification of the Laplace transform identities for the derived integrals.