Covariance matrix estimation under data-based loss

2020/12/22 by Anis M. Haddouche, Haddouche, Anis M., Dominique Fourdrinier +3
Engineering · Mathematics · #Advanced Statistical Methods and Models #Applications (stat.AP) #FOS: Computer and information sciences #FOS: Mathematics #Random Matrices and Applications #Sparse and Compressive Sensing Techniques #Statistics Theory (math.ST) #math.ST #stat.AP #stat.TH

paper · pdf · doi:10.48550/arxiv.2012.11920

arxiv created 2020/12/22 · openalex publication_date 2020/12/22 · arxiv updated 2020/12/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 Σ of Σ is assessed through the data-based loss tr(S⁺Σ (Σ^-1Σ - 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^-1Σ - 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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