The Annals of Statistics

Optimal rates of convergence for covariance matrix estimation

T. Tony Cai, Cun-Hui Zhang, and Harrison H. Zhou

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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.

Article information

Ann. Statist., Volume 38, Number 4 (2010), 2118-2144.

First available in Project Euclid: 11 July 2010

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Zentralblatt MATH identifier

Primary: 62H12: Estimation
Secondary: 62F12: Asymptotic properties of estimators 62G09: Resampling methods

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


Cai, T. Tony; Zhang, Cun-Hui; Zhou, Harrison H. Optimal rates of convergence for covariance matrix estimation. Ann. Statist. 38 (2010), no. 4, 2118--2144. doi:10.1214/09-AOS752.

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