Integral Representations and Approximations for Multivariate Gamma Distributions

Abstract

Let R be a p×p-correlation matrix with an “m-factorial” inverse R ⁻¹ = D − BB′ with diagonal D minimizing the rank m of B. A new $\left(m+1 \atop 2\right)$-variate integral representation is given for p-variate gamma distributions belonging to R, which is based on the above decomposition of R ⁻¹ without the restriction D > 0 required in former formulas. This extends the applicability of formulas with small m. For example, every p-variate gamma cdf can be computed by an at most $\left(p-1 \atop 2\right)$-variate integral if p = 3 or p = 4. Since computation is only feasible for small m, a given R is approximated by an m-factorial R ₀. The cdf belonging to R is approximated by the cdf associated with R ₀ and some additional correction terms with the deviations between R and R ₀.