A density-free approach to the matrix variate beta distribution

Abstract

Let $ S_1\sim W_k(n_1;\boldsymbol\Sigma)$ and $ S_2\sim W_k(n_2,\boldsymbol\Sigma)$ be independent Wishart matrices and $\boldsymbol\Sigma$ be p.d. Consider $ S= S_1+ S_2$ and define $ U=(S)ˆ{-½} S_1[( S)ˆ{-½}]$, where $( S)ˆ{½}$ is the unique lower triangular matrix with positive diagonal elements, such that $( S)ˆ{½}[( S)ˆ{½ }]'= S$, and $( S)ˆ{-½}$ is the inverse of $( S)ˆ{½}$. The joint distribution of the element of the symmetric matrix $ U$ is said to be `matrix variate beta' and is denoted by the symbol $B_k(n_½,n_2/2)$. Several interesting properties of this distribution are obtained in this paper.