Showing a useful result for Wisharts and Multivariate Beta random matrices

by Taylor   Last Updated August 14, 2019 00:19 AM

Let $\mathbf{A} \sim \text{Wishart}_m\left(k_a,\mathbf{V} \right)$ and $\mathbf{B} \sim \text{Wishart}_m\left(k_b,\mathbf{V} \right)$ be two full rank Wishart random matrices. Define $$ \mathbf{S} = \mathbf{A} + \mathbf{B} $$ and $$ \mathbf{U} = (\mathbf{T}^{-1})^{'}\mathbf{A}\mathbf{T}^{-1} $$ where $\mathbf{T}'\mathbf{T}$ is the Cholesky decomposition of $\mathbf{S}$. Show

  1. $\mathbf{S} \sim \text{Wishart}\left(k_1 + k_2, \mathbf{V} \right)$
  2. $\mathbf{U} \sim \text{Multivariate Beta}_m\left(\frac{k_1}{2}, \frac{k_2}{2}\right)$, and
  3. $\mathbf{S}$ is independent of $\mathbf{U}$.

I'm trying to show it using densities with respect to Lebesgue measure. Murihead's book goes through a lot of this stuff, but appeals to k-forms, which I'm not very comfortable with, and I'm trying to avoid at the moment. Apparently it's also true in the not-full-rank case (c.f. Uhlig 1994), but I'd like to tackle the simpler version first.

This book, so I'm working through it at the moment.



Related Questions


Updated November 06, 2018 15:19 PM

Updated May 22, 2018 17:19 PM

Updated September 08, 2018 21:19 PM

Updated February 11, 2018 14:19 PM

Updated August 31, 2018 11:19 AM