Analogues of multivariate beta (Dirichlet) distributions

Abstract

Let $ S\sim W_k(n,\boldsymbol\Sigma)$. Write $ W= S/\text{tr}\,\boldsymbol\Sigma ˆ - S$ and denote the distribution of $ W$ by $D_k(n,\boldsymbol\Sigma)$. Also if $p\sim B(a/2,b/2)$, and $p$ and $ W$ are independently distributed, denote the distribution of $p W$ by $D_k(a,b,n,\boldsymbol\Sigma)$. In this paper we study these two families of multivariate distribution and obtain various interesting properties concerning them.