Optimal rates of convergence for covariance matrix estimation

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

Research output: Contribution to journalArticlepeer-review

206 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)2118-2144
Number of pages27
JournalAnnals of Statistics
Issue number4
StatePublished - Aug 2010

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


  • Covariance matrix
  • Frobenius norm
  • Minimax lower bound
  • Operator norm
  • Optimal rate of convergence
  • Tapering

Fingerprint Dive into the research topics of 'Optimal rates of convergence for covariance matrix estimation'. Together they form a unique fingerprint.

Cite this