NON-CENTRAL MATRIX-VARIATE DIRICHLET DISTRIBUTION

Abstract

Let $X_{i}\sim W_{p}(n_{i},\Sigma,\Theta)$ where $\Theta= diag(\theta^{2}_{i},0,\ldots,0)$, $i=1,\ldots,r+1$. In this article the authors have derived the joint distribution of $U_i=C^{-1}X_{i}C'{}^{-1}$, $i=1,\ldots,r$ where $\sum_{i=1}^{r+1}X_{i}=CC'$ and $C$ is a lower triangular matrix. The joint distribution of $U_{1},\ldots,U_{r}$ is a non-central matrix-variate Dirichlet distribution. Several properties of this distribution such as marginal and conditional distributions, distribution of partial sums, moments and asymptotic results have also been studied.