In Gupta & Nagar's book *Matrix variate distributions*, the noncentral Beta type I(B) distribution with parameters $a$, $b$ and noncentrality parameter $\Theta$ is defined by $U={(S_1+S_2)}^{-\frac12}S_1{(S_1+S_2)}^{-\frac12}$ where $S_1 \sim W(2a, I, \Theta)$ is independent of $S_2\sim W(2b, I)$ (Wishart distributions), and the noncentral Beta type II(B) distribution is defined by $V={(S_2)}^{-\frac12}S_1{(S_2)}^{-\frac12}$ where $S_1$ and $S_2$ as above.

Then it is claimed that $V \sim U{(I-U)}^{-1}$. However my simulations show that this distributional equality does not hold. I've found that it holds when $\Theta = \textrm{diag}(\theta, \ldots, \theta)$. Otherwise I get $\det(V) \sim \det\bigl(U{(I-U)}^{-1}\bigr)$ and $\textrm{tr}(V) \sim \textrm{tr}\bigl(U{(I-U)}^{-1}\bigr)$ but not $V \sim U{(I-U)}^{-1}$.

Is it true that that $V \sim U{(I-U)}^{-1}$? That would mean I do something wrong in my simulations but I'm pretty sure my simulations are correct, and I'm rather wondering why the equality holds when $\Theta = \textrm{diag}(\theta, \ldots, \theta)$.

### Note

With my simulations again, I've found that $V \sim U{(I-U)}^{-1}$ if we define $U = S_1^\frac12{(S_1+S_2)}^{-1}S_1^\frac12$ and $V = S_1^\frac12 S_2^{-1}S_1^\frac12$. These are other proposed definitions of the matrix Beta distributions.

# Update

I think I misread. In fact the book is a bit confusing regarding this point. There are two Beta type I(B) distributions in the book: the one defined above and another one which was originally defined by Asoo. The book claims that $V \sim U{(I-U)}^{-1}$ when $U$ has the Asoo distribution, there's no such claim for the $U$ defined as above. So the real question is: why this equality holds when $\Theta = \textrm{diag}(\theta, \ldots, \theta)$?