The noncentral Wishart as an exponential family, and its moments

While the noncentral Wishart distribution is generally introduced as the distribution of the random symmetric matrix Y 1 ∗ Y 1 + ⋯ + Y n ∗ Y n where Y 1 , … , Y n are independent Gaussian rows in R k with the same covariance, the present paper starts from a slightly more general definition, following the extension of the chi-square distribution to the gamma distribution. We denote by γ ( p , a ; σ ) this general noncentral Wishart distribution: the real number p is called the shape parameter, the positive definite matrix σ of order k is called the shape parameter and the semi-positive definite matrix a of order k is such that the matrix ω = σ a σ is called the noncentrality parameter. This paper considers three problems: the derivation of an explicit formula for the expectation of tr ( X h 1 ) … tr ( X h m ) when X ∼ γ ( p , a , σ ) and h 1 , … , h m are arbitrary symmetric matrices of order k , the estimation of the parameters ( a , σ ) by a method different from that of Alam and Mitra [K. Alam, A. Mitra, On estimated the scale and noncentrality matrices of a Wishart distribution, Sankhyā, Series B 52 (1990) 133–143] and the determination of the set of acceptable p ’s as already done by Gindikin and Shanbag for the ordinary Wishart distribution γ ( p , 0 , σ ) .