Matrix Variate Kummer-dirichlet Distributions

(1.1) { Γ(α)Ψ(α,α−γ+1;ξ) }−1 exp(−ξv)v(1+v), v > 0, (1.2) respectively, where α > 0, β > 0, ξ > 0, −∞ < γ,λ < ∞, 1F1, and Ψ are confluent hypergeometric functions. These distributions are extensions of Gamma and Beta distributions, and for α < 1 (and certain values of λ and γ) yield bimodal distributions on finite and infinite ranges, respectively. These distributions are used (i) in the Bayesian analysis of queueing system where posterior distribution of certain basic parameters in M/M/∞ queueing system is Kummer-Gamma and (ii) in common value auctions where the posterior distribution of “value of a single good” is Kummer-Beta. For properties and applications of these distributions the reader is referred to Ng and Kotz [7], Armero and Bayarri [1], and Gordy [2].