Covariance matrix estimation under data-based loss
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In this paper, we consider the problem of estimating the p× p scale matrix Σ of a multivariate linear regression model Y=X β + 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 Σ̂ of Σ is assessed through the data-based loss tr(S^+Σ (Σ^-1Σ̂ - 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 a S, where a is a positive constant, which present smaller associated risk. Compared to the usual quadratic loss tr(Σ^-1Σ̂ - 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.
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