The randomization by Wishart laws and the Fisher information
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Consider the centered Gaussian vector X in ^n with covariance matrix Σ. Randomize Σ such that Σ^-1 has a Wishart distribution with shape parameter p>(n-1)/2 and mean pσ. We compute the density f_p,σ of X as well as the Fisher information I_p(σ) of the model (f_p,σ ) when σ is the parameter. For using the Cramér-Rao inequality, we also compute the inverse of I_p(σ). The important point of this note is the fact that this inverse is a linear combination of two simple operators on the space of symmetric matrices, namely (σ)(s)=σ s σ and (σ⊗σ)(s)=σ trace(σ s). The Fisher information itself is a linear combination (σ^-1) and σ^-1⊗σ^-1. Finally, by randomizing σ itself, we make explicit the minoration of the second moments of an estimator of σ by the Van Trees inequality: here again, linear combinations of (u) and u⊗ u appear in the results.
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