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Optimal rates of convergence for covariance matrix estimation. (English) Zbl 1202.62073
Summary: The 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. We establish the optimal rates of convergence for estimating the covariance matrix under both the operator norm and the 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.

MSC:
62H12 Estimation in multivariate analysis
62F12 Asymptotic properties of parametric estimators
62C20 Minimax procedures in statistical decision theory
65C60 Computational problems in statistics (MSC2010)
62G09 Nonparametric statistical resampling methods
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