Question

This is a cross-posting of a question I asked at CrossValidated. It hasn't generated much activity so I'm trying here:

Suppose $X\sim \operatorname{InvWishart}(\nu, \Sigma_0)$. I'm interested in the marginal distribution of the diagonal elements $\operatorname{diag}(X) = (x_{11}, \dots, x_{pp})$. There are a few simple results on the distribution of submatrices of $X$. From these I can figure that the marginal distribution of any single element on the diagonal is inverse gamma. But I've been unable to deduce the joint distribution. I suspect that I'm missing something simple; it seems like this "ought" to be known but I haven't been able to find/show it.

Answer 1

In the following reference

\textsc{Letac, G. and Weso{\l}owski, J.} (2000) 'An independence property for the product of GIG and gamma laws.' {\it Ann. Probab.} {\bf 28}, 1371-1383.

you will find at the end of the paper some remarks about the function $K(a,b,p)=\int_{P_n}\exp( -trace[ax+bx^{-1}])\det^{p-(n+1)/2}dx$ where $P_n$ is the cone of positive definite matrices where $a,b\in P_n$ and where $p>(n-1)/2.$ Up to some constants, the function $b\mapsto K(a,b,p)$ is basically the Laplace transform of $X^{-1}$ when $X$ is Wishart with parameters $p$ and $a.$ Taking $b$ as a diagonal matrix gives you the Laplace transform of the diagonal elements of the random matrix $X^{-1}.$

Answer 2

Hi Michael, do you have any progress on this topic? I had the same question, and don't want to start a new one, so just want to ask if you have any reference or maybe answer on this question.

As you said, the diagonal entries for Wishart distribution are chi-square distributed, and the y are inverse-gamma distributed for Inverse Wishart. Do you know the joint marginal distribution?