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Let $\Sigma$ be an $p\times p$ dimensional covariance matrix that is distributed Inverse Wishart with degrees of freedom $\nu$ and Prior scale matrix $\Psi$ such that we write $\Sigma \sim W^{-1}(\nu, \Psi)$. Also let $A$ be a $q \times p$ matrix. What is the distribution of $A \Sigma A^T$. Please use the parameterization of the Inverse Wishart from Wikipedia in your answer.

Is the answer just $A \Sigma A^T \sim W^{-1}(\nu, A\Psi A^T)$? For some reason, I am having trouble finding any references with this or any result for the Inverse Wishart (I see plenty for the Wishart).

In case it matters, what about for the special case where $q=p$ and $A$ is invertible?

jds
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  • This is a pretty good source: https://projecteuclid.org/euclid.lnms/1196285102#toc check out the whole chapter on Wisharts and then just use the relationship between inverse wisharts and wisharts – Taylor Jul 20 '19 at 04:36

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From the definition of the Wishart distribution, we have that if $g_i \sim \mathcal{N}_p(0, \Sigma)$, then $W = \sum_{i=1}^{\nu} g_ig_i^T$ is Wishart distributed $W \sim \mathcal{W}_p(\Sigma, \nu)$.

Since $Ag_i \sim \mathcal{N}_p(0, A\Sigma A^T)$, we have the following result: $$ AWA^T = \sum_{i=1}^{\nu} Ag_i(Ag_i)^T \sim \mathcal{W}_p(A\Sigma A^T, \nu)\,. $$

But what about inverse Wishart? That seems a little less trivial. We have that if $W \sim \mathcal{W}_p(\Sigma, \nu)$, $W^{-1} \sim \mathcal{W}^{-1}_p(\Sigma^{-1}, \nu)$. This implies that $(AWA^T)^{-1}\sim \mathcal{W}^{-1}_p((A\Sigma A^T)^{-1}, \nu)$.

If $A$ is invertible, we furthermore have $(AWA^T)^{-1} = (A^{-1})^TW^{-1}A^{-1}\sim \mathcal{W}^{-1}_p((A^{-1})^T\Sigma^{-1}A^{-1}, \nu)$.

Letting $B=(A^{-1})^T$, we have $BW^{-1}B^T\sim \mathcal{W}^{-1}_p(B\Sigma^{-1}B^T, \nu)$ if B is nonsingular, which is almost the required result.

I am not sure whether this can be extended to singular matrices $B$, but I hope that this helps (three years later).

Hudson Cooper
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