2022/11/25 by Gérard G. Letac, Letac, Gérard G.
Mathematics · #62F15 (secondary) #62H10 (primary) #FOS: Mathematics #Morphological variations and asymmetry #Point processes and geometric inequalities #Random Matrices and Applications #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2211.14137
openalex publication_date 2022/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider the centered Gaussian vector X in \Rn 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 fp,σ of X as well as the Fisher information Ip(σ) of the model (fp,σ ) when σ is the parameter. For using the Cramér-Rao inequality, we also compute the inverse of Ip(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)=σs σ and (σ⊗ σ)(s)=σ trace(σs). 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.