Proof of the conjectures of H. Uhlig on the singular multivariate beta and the Jacobian of a certain matrix transformation

Abstract

Uhlig proposes two conjectures. The first concerns the Jacobian of the transformation $Y = B \times B'$ where B is the matrix $m \times m$ and m and X, Y belong to the class of positive semidefinite matrices of the order of $m \times m$ of rank $n < m, S_{m,n^{\cdot}}^+$. The second is concerned with the singular multivariate Beta distribution. This article seeks to prove the two conjectures. The latter result is then extended to the case of the singular multivariate F distribution, and the respective density functions are located for the nonzero positive eigenvalues of the singular Beta and F matrices.