Covariance matrix estimation under data-based loss

In this paper, we consider the problem of estimating the $p\times p$ scale matrix $\mathrm{\Sigma }$ of a multivariate linear regression model $Y=X,\beta + $\mathcal{E}$,$whenthedistributionoftheobservedmatrix$Y$ belongs to a large class of elliptically symmetric distributions. After deriving the canonical form $(Z^T{U}^T{)}^T$ of this model, any estimator $\widehat{\mathrm{\Sigma }}$ of $\mathrm{\Sigma }$ is assessed through the data-based loss tr$(S^{+}\Sigma, (\Sigma^{-1}\hat{\Sigma} - I_p)^2 ),$where$S=U^T U$ is the sample covariance matrix and $S^{+}$ is its Moore-Penrose inverse. We provide alternative estimators to the usual estimators $aS$, where $a$ is a positive constant, which present smaller associated risk. Compared to the usual quadratic loss tr$({\mathrm{\Sigma }}^{-1}\widehat{\mathrm{\Sigma }}-I_p{)}^2$, we obtain a larger class of estimators and a wider class of elliptical distributions for which such an improvement occurs. A numerical study illustrates the theory.