Optimal rates of convergence for covariance matrix estimation

Details Optimal rates of convergence for covariance matrix estimation Optimal rates of convergence for covariance matrix estimation

2010-10-19

arxiv.org/abs/1010.3866v1

Annals of Statistics 2010, Vol. 38, No. 4, 2118-2144

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/09-AOS752 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of

Scientific paper

10.1214/09-AOS752

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.

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