Scale matrix estimation of an elliptically symmetric distribution in high and low dimensions

Anis M. Haddouche, Dominique Fourdrinier and Fatiha Mezoued

Journal of Multivariate Analysis, 2021, vol. 181, issue C

Abstract: The problem of estimating the scale matrix Σ in a multivariate additive model, with elliptical noise, is considered from a decision-theoretic point of view. As the natural estimators of the form Σˆa=aS (where S is the sample covariance matrix and a is a positive constant) perform poorly, we propose estimators of the general form Σˆa,G=a(S+SS+G(Z,S)), where S+ is the Moore–Penrose inverse of S and G(Z,S) is a correction matrix. We provide conditions on G(Z,S) such that Σˆa,G improves over Σˆa under the quadratic loss L(Σ,Σˆ)=tr(ΣˆΣ−1−Ip)2. We adopt a unified approach to the two cases where S is invertible and S is singular. To this end, a new Stein–Haff type identity and calculus on eigenstructure for S are developed. Our theory is illustrated with a large class of estimators which are orthogonally invariant.

Keywords: Elliptically symmetric distributions; High-dimensional statistics; Orthogonally invariant estimators; Quadratic loss; Stein–Haff type identities (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (1)

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DOI: 10.1016/j.jmva.2020.104680

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