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I am interested in calculating the expectation of the following random matrix: $$A=WX(X^TWX)^{-1},$$ where $W \sim W_p(n,I)$ is a $p\times p$ random Wishart matrix, and $X$ is a fixed $p\times m$ matrix, where $n>p>m$ (so that $X^TWX$ is reversible).

The question is, how can we calculate $\mathbb E[A]$? Any help or hints would be appreciated. Thank you!

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  • $\begingroup$ I believe there should be a neat result here. For example, it is straightforward to show that any such expectation (i.e. of a left- or right-inverse) is of the same form $M X (X^T M X)^{-1}$ for some matrix $M$, independent of the distribution of $W$. $\endgroup$
    – user114668
    Commented Mar 22, 2019 at 18:54
  • $\begingroup$ Thank you. Yes you're right, We expect this $M$ should be something equal to the Identity Matrix added by a small term, but we couldn't find out what this small term would be. $\endgroup$
    – user482401
    Commented Mar 22, 2019 at 19:07
  • $\begingroup$ Did you mean $m > p$ for the matrix $X^T W X$ to be invertible? $\endgroup$
    – user114668
    Commented Mar 22, 2019 at 19:51
  • $\begingroup$ Oh there's a typo. The size of $X$ should be $p\times m$. $\endgroup$
    – user482401
    Commented Mar 22, 2019 at 20:08

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