On the singular gamma, Wishart, and beta matrix-variate density functions

Abstract

When a $p\times p$ real positive definite matrix $S$ follows a Wishart or, more generally, a matrix-variate gamma distribution with shape parameter $\alpha$ and positive definite scale parameter matrix $B$, one can represent $S$ as $X{X}^{\prime }$ for some matrix $X$ of dimension $p\times q$. When $p>q$, $S$ has a singular distribution whose properties can be studied via the density function of $X$. It will be shown that when $X$ follows a matrix-variate extended Gaussian distribution, the density function of the resulting singular gamma distribution can be obtained by making use of successive transformations and their associated Jacobians. The singular Wishart distribution will then be obtained as a particular case. The marginal and conditional density functions resulting from an arbitrary partitioning of $X$ will be considered as well. The same technique will also be applied to the derivation of the density functions of real and complex singular type-1 and type-2 beta-distributed matrices. It so happens that the proposed approach, which is based on manipulations involving the wedge products of certain differential elements, generally proves more efficient than the intricate procedures that have hitherto been employed in the literature.