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 Y1∗Y1+⋯+Yn∗Yn where Y1,…,YnY1,…,Yn are independent Gaussian rows in RkRk 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)γ(p,a;σ) this general noncentral Wishart distribution: the real number pp is called the shape parameter, the positive definite matrix σσ of order kk is called the shape parameter and the semi-positive definite matrix aa of order kk is such that the matrix ω=σaσω=σaσ is called the noncentrality parameter. This paper considers three problems: the derivation of an explicit formula for the expectation of tr(Xh1)…tr(Xhm) when X∼γ(p,a,σ)X∼γ(p,a,σ) and h1,…,hmh1,…,hm are arbitrary symmetric matrices of order kk, the estimation of the parameters (a,σ)(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 pp’s as already done by Gindikin and Shanbag for the ordinary Wishart distribution γ(p,0,σ)γ(p,0,σ).