This is a reference request for the following result. Let $X$ be a random matrix following the matrix variate $t$-distribution $T_{p,m}(\nu, M, U, V)$ (as defined in Wikipedia). Then $$ \frac{\det(U)}{\det\bigl(U + (X-M)V^{-1}{(X-M)}'\bigr)} $$ has the same distribution as the product $B_1 \ldots B_m$ of $m$ independent random variables $B_i \sim \textrm{Beta}\left(\frac{k+1-i}{2}, \frac{p}{2}\right)$$B_i \sim \textrm{Beta}\left(\frac{\nu+m-i}{2}, \frac{p}{2}\right)$.
This result is checked by simulations in R below.
This result is given in Geisser's article Bayesian estimation in multivariate analysis but it is a bit hidden: it is given in a Bayesian context where a particular matrix variate $t$-distribution appears without name, and the proof is only sketched.
I'm looking for a reference which precisely states this result. It is not given in Gupta & Nagar's book Matrix variate distributions.
library(LaplacesDemon) # to simulate Wishart distribution
# simulates matrix normal N_{p,m}(M, U ⊗ V)
rmatrixnormal <- function(M, U, V){
Z <- matrix(rnorm(nrow(M)*ncol(M)), nrow(M), ncol(M))
M + t(chol(U)) %*% Z %*% chol(V)
}
# parameters of the matrix t-distribution
p <- 5; m <- 3
U <- rwishart(p, diag(p)) # arbitray U matrix
V <- rwishart(m, diag(m)) # arbitrary V matrix
Vinv <- solve(V) # inverse V
nu <- 6 # degrees of freedom
# simulations of matrix t-distribution
nsims <- 25000
sims <- numeric(nsims)
for(i in 1:nsims){
Psi <- rinvwishart(nunu+m-1, V)
X <- rmatrixnormal(matrix(0,p,m), U, Psi)
sims[i] <- det(U)/det(U + X %*% Vinv %*% t(X))
}
# simulations of product of m Beta distributions
k <- nu(nu+m-1)/2
sims2 <- rbeta(nsims, k, p/2)*rbeta(nsims, k-0.5, p/2)*rbeta(nsims, k-1, p/2)
# compare cumulative distribution functions
curve(ecdf(sims)(x), from=0, to=1)
curve(ecdf(sims2)(x), col="red", add=TRUE, lty=2, lwd=3)
