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Covariance matrix plays a central role in multivariate statistical analysis.Significant advances have been made recently on developing both theory andmethodology for estimating large covariance matrices. However, a minimax theoryhas yet been developed. In this paper we establish the optimal rates ofconvergence for estimating the covariance matrix under both the operator normand Frobenius norm. It is shown that optimal procedures under the two norms aredifferent and consequently matrix estimation under the operator norm isfundamentally different from vector estimation. The minimax upper bound isobtained by constructing a special class of tapering estimators and by studyingtheir risk properties. A key step in obtaining the optimal rate of convergenceis the derivation of the minimax lower bound. The technical analysis requiresnew ideas that are quite different from those used in the more conventionalfunction/sequence estimation problems.

Optimal rates of convergence for covariance matrix estimation