Scale matrix estimation under data-based loss in high and low dimensions

We consider the problem of estimating the scale matrix $\Sigma$ of the additif model $Y_{p\times n} = M + \mathcal{E}$, under a theoretical decision point of view. Here, $p$ is the number of variables, $n$ is the number of observations, $M$ is a matrix of unknown parameters with rank $q<p$ and $\mathcal {E}$ is a random noise, whose distribution is elliptically symmetric with covariance matrix proportional to $I_n \otimes \Sigma$\,. We deal with a canonical form of this model where $Y$ is decomposed in two matrices, namely, $Z_{q\times p}$ which summarizes the information contained in $M$, and $U_{m\times p}$, where $m=n-q$, which summarizes the sufficient information to estimate $\Sigma$. As the natural estimators of the form ${\hat {\Sigma}}_a=a\, S$ (where $S=U^{T}\,U$ and $a$ is a positive constant) perform poorly when $p >m$ (S non-invertible), we propose estimators of the form ${\hat{\Sigma}}_{a, G} = a\big( S+ S \, {S^{+}\,G(Z,S)}\big)$ where ${S^{+}}$ is the Moore-Penrose inverse of $S$ (which coincides with $S^{-1}$ when $S$ is invertible). We provide conditions on the correction matrix $SS^{+}{G(Z,S)}$ such that ${\hat {\Sigma}}_{a, G}$ improves over ${\hat {\Sigma}}_a$ under the data-based loss $L _S( \Sigma, \hat { \Sigma}) ={\rm tr} \big ( S^{+}\Sigma\,({\hat{\Sigma}} \, {\Sigma} ^ {- 1} - {I}_ {p} )^ {2}\big)$. We adopt a unified approach of the two cases where $S$ is invertible ($p \leq m$) and $S$ is non-invertible ($p>m$).