Marginal distribution of the diagonal of an inverse Wishart distributed matrix - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T20:47:47Z http://mathoverflow.net/feeds/question/63316 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63316/marginal-distribution-of-the-diagonal-of-an-inverse-wishart-distributed-matrix Marginal distribution of the diagonal of an inverse Wishart distributed matrix JMS 2011-04-28T17:44:58Z 2012-02-10T09:22:12Z <p>This is a cross-posting of a question I asked at <a href="http://stats.stackexchange.com/questions/10006/marginal-distribution-of-the-diagonal-of-an-inverse-wishart-distributed-matrix" rel="nofollow">CrossValidated</a>. It hasn't generated much activity so I'm trying here:</p> <p>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. </p> http://mathoverflow.net/questions/63316/marginal-distribution-of-the-diagonal-of-an-inverse-wishart-distributed-matrix/76150#76150 Answer by MPQ for Marginal distribution of the diagonal of an inverse Wishart distributed matrix MPQ 2011-09-22T19:18:51Z 2011-09-22T19:18:51Z <p>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.</p> <p>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?</p> http://mathoverflow.net/questions/63316/marginal-distribution-of-the-diagonal-of-an-inverse-wishart-distributed-matrix/84577#84577 Answer by Gerard Letac for Marginal distribution of the diagonal of an inverse Wishart distributed matrix Gerard Letac 2011-12-30T06:40:30Z 2011-12-30T06:40:30Z <p>In the following reference</p> <p>\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.</p> <p>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 $X^{-1}.$ </p>