Singular Wishart and multivariate beta distributions

In this article, we consider the case when the number of observations n is less than the dimension p of the random vectors which are assumed to be independent and identically distributed as normal with nonsingular covariance matrix. The central and noncentral distributions of the singular Wishart matrix S = XX', where X is the p x n matrix of observations are derived with respect to Lebesgue measure. Properties of this distribution are given. When the covariance matrix is singular, pseudo singular Wishart distribution is also derived. The result is extended to any distribution of the type f (XX') for the central case. Singular multivariate beta distributions with respect to Lebesgue measure are also given.