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.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 2118-2144 |
| Number of pages | 27 |
| Journal | Annals of Statistics |
| Volume | 38 |
| Issue number | 4 |
| DOIs | |
| State | Published - Aug 1 2010 |
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All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Statistics, Probability and Uncertainty
Keywords
- Covariance matrix
- Frobenius norm
- Minimax lower bound
- Operator norm
- Optimal rate of convergence
- Tapering
Cite this
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Optimal rates of convergence for covariance matrix estimation. / Cai, T. Tony; Zhang, Cun Hui; Zhou, Harrison H.
In: Annals of Statistics, Vol. 38, No. 4, 01.08.2010, p. 2118-2144.Research output: Contribution to journal › Article
TY - JOUR
T1 - Optimal rates of convergence for covariance matrix estimation
AU - Cai, T. Tony
AU - Zhang, Cun Hui
AU - Zhou, Harrison H.
PY - 2010/8/1
Y1 - 2010/8/1
N2 - 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.
AB - 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.
KW - Covariance matrix
KW - Frobenius norm
KW - Minimax lower bound
KW - Operator norm
KW - Optimal rate of convergence
KW - Tapering
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UR - http://www.scopus.com/inward/citedby.url?scp=77955132618&partnerID=8YFLogxK
U2 - 10.1214/09-AOS752
DO - 10.1214/09-AOS752
M3 - Article
AN - SCOPUS:77955132618
VL - 38
SP - 2118
EP - 2144
JO - Annals of Statistics
JF - Annals of Statistics
SN - 0090-5364
IS - 4
ER -