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We consider the problem of estimating the scale matrix $\Sigma$ of theadditif model $Y_{p\times n} = M + \mathcal{E}$, under a theoretical decisionpoint of view. Here, $ p $ is the number of variables, $ n$ is the number ofobservations, $ M $ is a matrix of unknown parameters with rank $q<p$ and $\mathcal {E}$ is a random noise, whose distribution is elliptically symmetricwith covariance matrix proportional to $ I_n \otimes \Sigma $\,. We deal with acanonical 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 toestimate $ \Sigma $. As the natural estimators of the form ${\hat{\Sigma}}_a=a\, S$ (where $ S=U^{T}\,U$ and $a$ is a positive constant) performpoorly 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 $S$ isinvertible). We provide conditions on the correction matrix $SS^{+}{G(Z,S)}$such that ${\hat {\Sigma}}_{a, G}$ improves over ${\hat {\Sigma}}_a$ under thedata-based loss $L _S( \Sigma, \hat { \Sigma}) ={\rm tr} \big (S^{+}\Sigma\,({\hat{\Sigma}} \, {\Sigma} ^ {- 1} - {I}_ {p} )^ {2}\big) $. Weadopt a unified approach of the two cases where $ S$ is invertible ($p \leq m$)and $ S$ is non-invertible ($p>m$).

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