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

JOURNAL OF MULTIVARIATE ANALYSIS

卷:181

Article

Haddouche, Anis M.^1,2  Fourdrinier, Dominique²  Mezoued, Fatiha¹ 

[1] LAMOPS, Ecole Natl Super & Stat & Econ Appl ENSSEA, Tipasa, Algeria

[2] Univ Normandie, UNIROUEN, UNIHAVRE, INSA Rouen,LITIS, Ave Univ BP 12, F-76801 St Etienne Du Rouvray, France

关键词: Elliptically symmetric distributions;    High-dimensional statistics;    Orthogonally invariant estimators;    Quadratic loss;    Stein-Haff type identities;   

DOI  :  10.1016/j.jmva.2020.104680

来源: Elsevier

【 摘 要 】

The problem of estimating the scale matrix Sigma in a multivariate additive model, with elliptical noise, is considered from a decision-theoretic point of view. As the natural estimators of the form Sigma a = a S (where S is the sample covariance matrix and a is a positive constant) perform poorly, we propose estimators of the general form Sigma a,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 Sigma a,G improves over Sigma a under the quadratic loss L(Sigma, Sigma)= tr(Sigma Sigma-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. (c) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free