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Given a sequence of real Wishart matrices $W_1 , \cdots , W_k \sim \mathcal{W}_m(n,\Sigma)$ where $\Sigma$ is a singular matrix. Are there good estimates for the degrees of freedom?

The MLE for $\Sigma$ is $\frac{1}{k\cdot n-1}\sum_{i=1}^k W_i$ since this is the MLE if we had access to the samples which gave rise to the observed Wishart matrices.

The only estimate I have been able to come up for the degrees of freedom is to take a random vector $Y$ and consider $\frac{Y^T W Y}{Y^T (\frac{1}{k}\sum_{i=1}^k W_i) Y}$. This should be approximately $\chi^2_n /n$ according to theorem 3.2.8 in Muirhead's Aspects of multivariate statistical theory, estimating $n$ now comes down to estimating the shape parameter of the $\chi^2_n /n$.

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