Optimal Rates of Convergence for Covariance Matrix Estimation
Tony Cai, Cun-Hui Zhang, and Harrison H. Zhou
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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.
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