Proof of the conjectures of H. Uhlig on the singular multivariate beta and the Jacobian of a certain matrix transformation
关于奇异多元 Beta 分布与矩阵变换雅可比的 Uhlig 猜想证明
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…