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Integral Representations and Approximations for Multivariate Gamma Distributions
Authors:T Royen
Institution:(1) Fachhochschule Bingen, University of Applied Sciences, Berlinstrasse 109, 55411 Bingen, Germany
Abstract:Let R be a p×p-correlation matrix with an “m-factorial” inverse R −1 = DBB′ 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.
Keywords:Multivariate gamma distribution  Multivariate chi-square distribution  Multivariate Rayleigh-distribution  Approximation for positive definite matrices            m-factorial matrices
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