Noncentral matrix beta distributions of type I and II

Question

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 $\Theta = \textrm{diag}(\theta, \ldots, \theta)$?

Answer 1

The density of $U$ is (see Gupta & Nagar page 189) $$ f(U) = C_f {\det(U)}^{a - \frac12(p+1)} {\det(I_p-U)}^{b - \frac12(p+1)} \\ \int_{S>0} {\det(S)}^{a+b-\frac12(p+1)} \exp\left(-\mathrm{tr}\left(\frac{S}{2}\right)\right) {}_0\!F_1\left(a, \frac{1}{4}\Theta_1 S^\frac12 US^\frac12\right) \mathrm{d}S \\ = K_f {\det(U)}^{a - \frac12(p+1)} {\det(I_p-U)}^{b - \frac12(p+1)} \\ \int_{S>0} {\det(S)}^{a+b-\frac12(p+1)} \exp\bigl(-\mathrm{tr}(S)\bigr) {}_0\!F_1\left(a, \frac{1}{2}\Theta_1 S^\frac12 US^\frac12\right) \mathrm{d}S $$ (where $C_f$ and $K_f$ are constants).

The density of $V$ is (see Gupta & Nagar page 192) $$ g(V) = C_g {\det(V)}^{a - \frac12(p+1)} {\det(I+V)}^{\frac12 (a+b)} {}_1\!F_1\left(a+b, a, \frac12 \Theta_1 V{(I+V)}^{-1}\right). $$ Make the change of variables $U = V{(I_p+V)}^{-1}$, so $V = U{(I_p-U)}^{-1}$ and $I_p + V = {(I_p-U)}^{-1}$. The Jacobian is ${\det(I_p -U)}^{-(p+1)}$. We find that the density of $U$ is $$ h(U) = C_g {\det(U)}^{a - \frac12(p+1)} {\det(I_p-U)}^{b - \frac12(p+1)} {}_1\!F_1\left(a+b, a, \frac12 \Theta_1 U\right). $$

The integral in $f(U)$ does not simplify to something nice. Hence the density $f(U)$ and $h(U)$ are not the same.

But when $\Theta_1$ is scalar, by integral involving hypergeometric function of matrix argument, $$ f(U) = K_f {\det(U)}^{a - \frac12(p+1)} {\det(I_p-U)}^{b - \frac12(p+1)} \\ \int_{S>0} {\det(S)}^{a+b-\frac12(p+1)} \exp\bigl(-\mathrm{tr}(S)\bigr) {}_0\!F_1\left(a, \frac{1}{2}\Theta_1 S U\right) \mathrm{d}S \\ = K_f\Gamma_p(a+b) {\det(U)}^{a - \frac12(p+1)} {\det(I_p-U)}^{b - \frac12(p+1)} {}_1\!F_1\left(a+b, a, \frac12 \Theta_1 U\right), $$ the same as $h(U)$.