Optimal rates of convergence for covariance matrix estimation

TL;DR

This paper establishes the optimal convergence rates for covariance matrix estimation under operator and Frobenius norms, highlighting the fundamental differences from vector estimation and introducing tapering estimators.

Contribution

It develops the first minimax theory for covariance matrix estimation, providing optimal rates and new technical methods for deriving lower bounds.

Findings

Optimal rates differ under operator and Frobenius norms.

Tapering estimators achieve minimax upper bounds.

New techniques for minimax lower bounds in matrix estimation.

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.