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 ($p>m$).
Categories: math.ST stat.AP stat.TH