Statistics Papers

Document Type

Journal Article

Date of this Version

2010

Publication Source

The Annals of Statistics

Volume

38

Issue

4

Start Page

2118

Last Page

2144

DOI

10.1214/09-AOS752

Abstract

Covariance matrix plays a central role in multivariate statistical analysis. Significant advances have been made recently on developing both theory and methodology for estimating large covariance matrices. However, a minimax theory has yet been developed. In this paper we establish the optimal rates of convergence for estimating the covariance matrix under both the operator norm and Frobenius norm. It is shown that optimal procedures under the two norms are different and consequently matrix estimation under the operator norm is fundamentally different from vector estimation. The minimax upper bound is obtained by constructing a special class of tapering estimators and by studying their risk properties. A key step in obtaining the optimal rate of convergence is the derivation of the minimax lower bound. The technical analysis requires new ideas that are quite different from those used in the more conventional function/sequence estimation problems.

Keywords

covariance matrix, Frobenius norm, minimax lower bound, operator norm, optimal rate of convergence, tapering

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Date Posted: 27 November 2017

This document has been peer reviewed.