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I have a question about the inverse Wishart matrix. In my understanding, consider $\mathbf H$ is a $n\times n$ matrix with each elements are complex Gaussian with zero mean unit variance. Then $\mathbf H^* \mathbf H$ ($^*$ is Hermitian transpose) should be Wishart distribution $W_n(n, \mathbf I_n)$ and $(\mathbf H^* \mathbf H)^{-1}$ is inverse Wishart distribution $W^{-1}(\mathbf I_n, n)$.

I'm wondering how to find the order statistics distribution (or joint density function or marginal distribution) of the diagonal terms of the matrix $(\mathbf H^* \mathbf H)^{-1}$. I found that the marginal distribution of diagonal terms is inverse gamma (still try to derive it by myself). I still need to found the joint density function or ordered distribution.

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  • $\begingroup$ I'm pretty sure there's no closed-form expression. Of course, if your matrix is small you could just calculate this distribution by integration over the nondiagonal elements. $\endgroup$ Oct 16, 2015 at 6:22
  • $\begingroup$ I need to find the case for general size matrix, so I think it's not possible to integrate over all nondiagonal elements... There are some properties for Wishart matrix, but it seems only few can be applied to inverse Wishart $\endgroup$
    – YSW
    Oct 19, 2015 at 3:41

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