The randomization by Wishart laws and the Fisher information

Consider the centered Gaussian vector $X$ in ${\mathbb{R}}^n$ with covariance matrix $\mathrm{\Sigma }\mathrm{.}$ Randomize $\mathrm{\Sigma }$ such that ${\mathrm{\Sigma }}^{-1}$ has a Wishart distribution with shape parameter $p>(n-1)\mathrm{/}2$ and mean $p\sigma \mathrm{.}$ We compute the density $f_{p,\sigma }$ of $X$ as well as the Fisher information $I_p(\sigma )$ of the model $(f_{p,\sigma })$ when $\sigma$ is the parameter. For using the Cramér-Rao inequality, we also compute the inverse of $I_p(\sigma )$. 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 $\mathrm{\P }(\sigma )(s)=\sigma s\sigma$ and $(\sigma \otimes \sigma )(s)=\sigma \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(\sigma s)$. The Fisher information itself is a linear combination $\mathrm{\P }({\sigma }^{-1})$ and ${\sigma }^{-1}\otimes {\sigma }^{-1}\mathrm{.}$ Finally, by randomizing $\sigma$ itself, we make explicit the minoration of the second moments of an estimator of $\sigma$ by the Van Trees inequality: here again, linear combinations of $\mathrm{\P }(u)$ and $u\otimes u$ appear in the results.