Optimal rates of convergence for covariance matrix estimation

Annals of Statistics

Volumn 38, Issue 4, 2010, Pages 2118-2144

  Cai, T Tony ^a   Zhang, Cun Hui ^c   Zhou, Harrison H ^b  

^a UNIVERSITY OF PENNSYLVANIA   (United States)

^b YALE UNIVERSITY   (United States)

^c University Piscataway   (United States)

Author keywords

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

Indexed keywords

EID: 77955132618     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/09-AOS752     Document Type: Article

References (21)

1. ASSOUAD, P. (1983). Deux remarques sur l'estimation. C. R. Acad. Sci. Paris Sér. I Math. 296 1021-1024. MR0777600
2. BICKEL, P. J. and LEVINA, E. (2008a). Regularized estimation of large covariance matrices. Ann. Statist. 36 199-227. MR2387969
3. BICKEL, P. J. and LEVINA, E. (2008b). Covariance regularization by thresholding. Ann. Statist. 36 2577-2604. MR2485008
4. CSISZÁR, I. (1967). Information-type measures of difference of probability distributions and indirect observation. Studia Sci. Math. Hungar. 2 229-318. MR0219345
5. EL KAROUI, N. (2008). Operator norm consistent estimation of large dimensional sparse covariance matrices. Ann. Statist. 36 2717-2756. MR2485011
6. GOLUB, G. H. and VAN LOAN, C. F. (1983). Matrix Computations. John Hopkins Univ. Press, Baltimore. MR0733103
7. FAN, J., FAN, Y. and LV, J. (2008). High dimensional covariance matrix estimation using a factor model. J. Econometrics 147 186-197. MR2472991
8. FURRER, R. and BENGTSSON, T. (2007). Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants. J. Multivariate Anal. 98 227-255. MR2301751
9. HUANG, J., LIU, N., POURAHMADI, M. and LIU, L. (2006). Covariance matrix selection and estimation via penalised normal likelihood. Biometrika 93 85-98. MR2277742
10. JOHNSTONE, I. M. (2001). On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 295-327. MR1863961
11. JOHNSTONE, I. M. and LU, A. Y. (2009). On consistency and sparsity for principal components analysis in high dimensions. J. Amer. Statist. Assoc. 104 682-693.
12. LAM, C. and FAN, J. (2007). Sparsistency and rates of convergence in large covariance matrices estimation. Technical report, Princeton Univ.
13. MUIRHEAD, R. J. (1987). Developments in eigenvalue estimation. In Advances in Multivariate Statistical Analysis (A. K. Gupta, ed.) 277-288. Reidel, Dordrecht. MR0920436
14. 1-penalized log-determinant divergence. Technical report, Univ. California, Berkeley.
15. ROTHMAN, A. J., BICKEL, P. J., LEVINA, E. and ZHU, J. (2008). Sparse permutation invariant covariance estimation. Electron. J. Stat. 2 494-515. MR2417391
16. RUDELSON, M. and VERSHYNIN, R. (2007). Sampling from large matrices: An approach through geometric functional analysis. J. ACM 54 Art. 21, 19 pp. (electronic). MR2351844
17. SAULIS, L. and STATULEVIČIUS, V. A. (1991). Limit Theorems for Large Deviations. Springer, Berlin.
18. WU, W. B. and POURAHMADI, M. (2009). Banding sample covariance matrices of stationary processes. Statist. Sinica 19 1755-1768.
19. YU, B. (1997). Assouad, Fano and Le Cam. In Festschrift for Lucien Le Cam (D. Pollard, E. Torgersen and G. Yang, eds.) 423-435. Springer, Berlin. MR1462963
20. ZHANG, C.-H. and HUANG, J. (2008). The sparsity and bias of the Lasso selection in highdimensional linear regression. Ann. Statist. 36 1567-1594. MR2435448
21. ZOU, H., HASTIE, T. and TIBSHIRANI, R. (2006). Sparse principal components analysis. J. Comput. Graph. Statist. 15 265-286. MR2252527