Estimation of two high-dimensional covariance matrices and the spectrum of their ratio

JOURNAL OF MULTIVARIATE ANALYSIS

卷:168

Article

Wen, Jun¹ 

[1] Natl Univ Singapore, Dept Stat & Appl Probabil, Singapore 117546, Singapore

关键词: Covariance matrix estimation;    High-dimensional asymptotics;    Marcenko-Pastur equation;    Random matrix theory;    Spectrum estimation;    Two-sample problem;   

DOI  :  10.1016/j.jmva.2018.06.008

来源: Elsevier

【 摘 要 】

Let S-p,S-1, S-p,S-2 be two independent p x p sample covariance matrices with degrees of freedom n(1) and n(2), respectively, whose corresponding population covariance matrices are Sigma(p,1) and Sigma(p,2), respectively. Knowing S-p,S-1, S-p,S-2, this article proposes a class of estimators for the spectrum (eigenvalues) of the matrix Sigma(p,2) Sigma(-1)(p,1) as well as the pair of the whole matrices (Sigma(p,1), Sigma(p,2)). The estimators are created based on Random Matrix Theory. Under mild conditions, our estimator for the spectrum of Sigma(p,2)Sigma(-1)(p,1) is shown to be weakly consistent and the estimator for (Sigma(p,1), Sigma(p,2)) is shown to be optimal in the sense of minimizing the asymptotic loss within the class of equivariant estimators as n(1), n(2), p -> infinity with p/n(1) -> c(1) is an element of(0, 1), P/n(2) -> c(2) is an element of(0, 1) boolean OR (1, infinity). Also, our estimators are easy to implement. Even when p is 1000, our estimators can be computed in seconds using a personal laptop. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free