On Bayesian inference for generalized multivariate gamma distribution

In this paper we define a generalized multivariate gamma (MG) distribution and develop various properties of this distribution. Then we consider a Bayesian decision theoretic approach to develop the inference technique for the related scale matrix ΣΣ. We show that maximum posteriori (MAP) estimate is a Bayes estimator. We also develop the testing problem for ΣΣ using a Bayes factor. This approach provides a mathematically closed form solution for ΣΣ. The only other approach to Bayesian inference for the MG distribution is given in Tsionas (2004), which is based on Markov Chain Monte Carlo (MCMC) technique. The Tsionas (2004) technique involves a costly matrix inversion whose computational complexity increases in cubic order, hence making inference infeasible for ΣΣ, for large dimensions. In this paper, we provide an elegant closed form Bayes factor for ΣΣ.