Covariance matrix estimation under data-based loss

Abstract

In this paper, we consider the problem of estimating the $p\times p$ scale matrix $\Sigma$ of a multivariate linear regression model $Y=X\beta +\mathcal{E}$ when the distribution of the observed matrix $Y$ belongs to a large class of elliptically symmetric distributions. After deriving the canonical form $(Z^T{U}^T{)}^T$ of this model, any estimator $\hat{\Sigma }$ of $\Sigma$ 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$({\Sigma }^{-1}\hat{\Sigma }-I_p{)}^2$, 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.