{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,15]],"date-time":"2026-10-15T22:07:55Z","timestamp":1792102075051,"version":"4.4.0"},"reference-count":36,"publisher":"Institute of Mathematical Statistics","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ann. Statist."],"published-print":{"date-parts":[[2008,12,1]]},"abstract":"<jats:p>This paper considers regularizing a covariance matrix of p variables estimated from n observations, by hard thresholding. We show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense, the variables are Gaussian or sub-Gaussian, and (log\u2009p)\/n\u21920, and obtain explicit rates. The results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity. We discuss an intuitive resampling scheme for threshold selection and prove a general cross-validation result that justifies this approach. We also compare thresholding to other covariance estimators in simulations and on an example from climate data.<\/jats:p>","DOI":"10.1214\/08-aos600","type":"journal-article","created":{"date-parts":[[2009,1,5]],"date-time":"2009-01-05T09:20:00Z","timestamp":1231147200000},"page":"2577-2604","source":"Crossref","is-referenced-by-count":895,"title":["Covariance regularization by thresholding"],"prefix":"10.1214","volume":"36","author":[{"given":"Peter J.","family":"Bickel","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Elizaveta","family":"Levina","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"1","doi-asserted-by":"publisher","unstructured":"[1] Abramovich, F., Benjamini, Y., Donoho, D. and Johnstone, I. (2006). Adapting to unknown sparsity by controlling the false discovery rate. <i>Ann. Statist.<\/i> <b>34<\/b> 584\u2013653.","DOI":"10.1214\/009053606000000074"},{"key":"2","doi-asserted-by":"publisher","unstructured":"[2] Bickel, P. J. and Levina, E. (2004). Some theory for Fisher\u2019s linear discriminant function, \u201cnaive Bayes,\u201d and some alternatives when there are many more variables than observations. <i>Bernoulli<\/i> <b>10<\/b> 989\u20131010.","DOI":"10.3150\/bj\/1106314847"},{"key":"3","doi-asserted-by":"publisher","unstructured":"[3] Bickel, P. J. and Levina, E. (2008). Regularized estimation of large covariance matrices. <i>Ann. Statist.<\/i> <b>36<\/b> 199\u2013227.","DOI":"10.1214\/009053607000000758"},{"key":"4","unstructured":"[4] Bickel, P. J., Ritov, Y. and Zakai, A. (2006). Some theory for generalized boosting algorithms. <i>J. Mach. Learn. Res.<\/i> <b>7<\/b> 705\u2013732."},{"key":"5","doi-asserted-by":"publisher","unstructured":"[5] d\u2019Aspremont, A., Banerjee, O. and El Ghaoui, L. (2007). First-order methods for sparse covariance selection. <i>SIAM J. Matrix Anal. Appl.<\/i> <b>30<\/b> 56\u201366.","DOI":"10.1137\/060670985"},{"key":"6","doi-asserted-by":"crossref","unstructured":"[6] d\u2019Aspremont, A., El Ghaoui, L., Jordan, M. I. and Lanckriet, G. R. G. (2007). A direct formulation for sparse PCA using semidefinite programming. <i>SIAM Rev.<\/i> <b>49<\/b> 434\u2013448.","DOI":"10.1137\/050645506"},{"key":"7","doi-asserted-by":"publisher","unstructured":"[7] Dey, D. K. and Srinivasan, C. (1985). Estimation of a covariance matrix under Stein\u2019s loss. <i>Ann. Statist.<\/i> <b>13<\/b> 1581\u20131591.","DOI":"10.1214\/aos\/1176349756"},{"key":"8","doi-asserted-by":"publisher","unstructured":"[8] Donoho, D. L. and Johnstone, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage. <i>Biometrika<\/i> <b>81<\/b> 425\u2013455.","DOI":"10.1093\/biomet\/81.3.425"},{"key":"9","doi-asserted-by":"publisher","unstructured":"[9] Dudoit, S. and van der Laan, M. J. (2005). Asymptotics of cross-validated risk estimation in estimator selection and performance assessment. <i>Statist. Methodol.<\/i> <b>2<\/b> 131\u2013154.","DOI":"10.1016\/j.stamet.2005.02.003"},{"key":"12","doi-asserted-by":"crossref","unstructured":"[12] El Karoui, N. (2007c). Tracy\u2013Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices. <i>Ann. Probab.<\/i> <b>35<\/b> 663\u2013714.","DOI":"10.1214\/009117906000000917"},{"key":"15","doi-asserted-by":"publisher","unstructured":"[15] Fan, J. and Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. <i>J. Amer. Statist. Assoc.<\/i> <b>96<\/b> 1348\u20131360.","DOI":"10.1198\/016214501753382273"},{"key":"16","doi-asserted-by":"crossref","unstructured":"[16] Friedman, J., Hastie, T. and Tibshirani, R. (2008). Sparse inverse covariance estimation with the graphical lasso. <i>Biostatistics<\/i> <b>9<\/b> 432\u2013441.","DOI":"10.1093\/biostatistics\/kxm045"},{"key":"17","doi-asserted-by":"publisher","unstructured":"[17] Furrer, R. and Bengtsson, T. (2007). Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants. <i>J. Multivariate Anal.<\/i> <b>98<\/b> 227\u2013255.","DOI":"10.1016\/j.jmva.2006.08.003"},{"key":"20","doi-asserted-by":"publisher","unstructured":"[20] Haff, L. R. (1980). Empirical Bayes estimation of the multivariate normal covariance matrix. <i>Ann. Statist.<\/i> <b>8<\/b> 586\u2013597.","DOI":"10.1214\/aos\/1176345010"},{"key":"21","doi-asserted-by":"publisher","unstructured":"[21] Huang, J., Liu, N., Pourahmadi, M. and Liu, L. (2006). Covariance matrix selection and estimation via penalised normal likelihood. <i>Biometrika<\/i> <b>93<\/b> 85\u201398.","DOI":"10.1093\/biomet\/93.1.85"},{"key":"22","doi-asserted-by":"publisher","unstructured":"[22] Johnstone, I. and Silverman, B. (2005). Empirical Bayes selection of wavelet thresholds. <i>Ann. Statist.<\/i> <b>33<\/b> 1700\u20131752.","DOI":"10.1214\/009053605000000345"},{"key":"23","doi-asserted-by":"publisher","unstructured":"[23] Johnstone, I. M. (2001). On the distribution of the largest eigenvalue in principal components analysis. <i>Ann. Statist.<\/i> <b>29<\/b> 295\u2013327.","DOI":"10.1214\/aos\/1009210544"},{"key":"26","doi-asserted-by":"publisher","unstructured":"[26] Ledoit, O. and Wolf, M. (2003). A well-conditioned estimator for large-dimensional covariance matrices. <i>J. Multivariate Anal.<\/i> <b>88<\/b> 365\u2013411.","DOI":"10.1016\/S0047-259X(03)00096-4"},{"key":"28","doi-asserted-by":"publisher","unstructured":"[28] Levina, E., Rothman, A. J. and Zhu, J. (2008). Sparse estimation of large covariance matrices via a nested Lasso penalty. <i>Ann. Appl. Statist.<\/i> <b>2<\/b> 245\u2013263.","DOI":"10.1214\/07-AOAS139"},{"key":"29","doi-asserted-by":"crossref","unstructured":"[29] Mar\u010denko, V. A. and Pastur, L. A. (1967). Distributions of eigenvalues of some sets of random matrices. <i>Math. USSR-Sb<\/i> <b>1<\/b> 507\u2013536.","DOI":"10.1070\/SM1967v001n04ABEH001994"},{"key":"30","unstructured":"[30] Paul, D. (2007). Asymptotics of the leading sample eigenvalues for a spiked covariance model. <i>Statist. Sinica<\/i> <b>17<\/b> 1617\u20131642."},{"key":"31","doi-asserted-by":"publisher","unstructured":"[31] Rothman, A. J., Bickel, P. J., Levina, E. and Zhu, J. (2008). Sparse permutation invariant covariance estimation. <i>Electron. J. Stat.<\/i> <b>2<\/b> 494\u2013515.","DOI":"10.1214\/08-EJS176"},{"key":"34","doi-asserted-by":"publisher","unstructured":"[34] Wu, W. B. and Pourahmadi, M. (2003). Nonparametric estimation of large covariance matrices of longitudinal data. <i>Biometrika<\/i> <b>90<\/b> 831\u2013844.","DOI":"10.1093\/biomet\/90.4.831"},{"key":"35","doi-asserted-by":"publisher","unstructured":"[35] Yuan, M. and Lin, Y. (2007). Model selection and estimation in the Gaussian graphical model. <i>Biometrika<\/i> <b>94<\/b> 19\u201335.","DOI":"10.1093\/biomet\/asm018"},{"key":"36","doi-asserted-by":"publisher","unstructured":"[36] Zou, H., Hastie, T. and Tibshirani, R. (2006). Sparse principal components analysis. <i>J. Comput. Graph. Statist.<\/i> <b>15<\/b> 265\u2013286.","DOI":"10.1198\/106186006X113430"},{"key":"10","doi-asserted-by":"publisher","unstructured":"[10] El Karoui, N. (2007a). Operator norm consistent estimation of large-dimensional sparse covariance matrices. <i>Ann. Statist.<\/i> To appear.","DOI":"10.1214\/07-AOS559"},{"key":"11","doi-asserted-by":"publisher","unstructured":"[11] El Karoui, N. (2007b). Spectrum estimation for large-dimensional covariance matrices using random matrix theory. <i>Ann. Statist.<\/i> To appear.","DOI":"10.1214\/07-AOS581"},{"key":"13","doi-asserted-by":"publisher","unstructured":"[13] Fan, J., Fan, Y. and Lv, J. (2008). High-dimensional covariance matrix estimation using a factor model. <i>J. Econometrics<\/i>. To appear.","DOI":"10.1016\/j.jeconom.2008.09.017"},{"key":"14","unstructured":"[14] Fan, J., Feng, Y. and Wu, Y. (2007). Network exploration via the adaptive LASSO and SCAD penalties. Unpublished manuscript."},{"key":"18","unstructured":"[18] Golub, G. H. and Van Loan, C. F. (1989). <i>Matrix Computations<\/i>, 2nd ed. Johns Hopkins Univ. Press, Baltimore, MD."},{"key":"19","doi-asserted-by":"crossref","unstructured":"[19] Gyorfi, L., Kohler, M., Krzyzak, A. and Walk, H. (2002). <i>A Distribution-Free Theory of Nonparametric Regression<\/i>. Springer, New York.","DOI":"10.1007\/b97848"},{"key":"24","unstructured":"[24] Johnstone, I. M. and Lu, A. Y. (2004). Sparse principal components analysis. Unpublished manuscript."},{"key":"25","unstructured":"[25] Lam, C. and Fan, J. (2007). Sparsistency and rates of convergence in large covariance matrices estimation. Manuscript."},{"key":"27","unstructured":"[27] Ledoux, M. (2001). <i>The Concentration of Measure Phenomenon<\/i>. Amer. Math. Soc., Providence, RI."},{"key":"32","doi-asserted-by":"crossref","unstructured":"[32] Saulis, L. and Statulevi\u010dius, V. A. (1991). <i>Limit Theorems for Large Deviations<\/i>. Kluwer, Dordrecht.","DOI":"10.1007\/978-94-011-3530-6"},{"key":"33","doi-asserted-by":"crossref","unstructured":"[33] van der Vaart, A. W. and Wellner, J. A. (1996). <i>Weak Convergence and Empirical Processes: With Applications to Statistics<\/i>. Springer, New York.","DOI":"10.1007\/978-1-4757-2545-2"}],"container-title":["The Annals of Statistics"],"original-title":[],"link":[{"URL":"http:\/\/projecteuclid.org\/download\/pdfview_1\/euclid.aos\/1231165180","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T23:16:20Z","timestamp":1791587780000},"score":1,"resource":{"primary":{"URL":"https:\/\/projecteuclid.org\/journals\/annals-of-statistics\/volume-36\/issue-6\/Covariance-regularization-by-thresholding\/10.1214\/08-AOS600.full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,12,1]]},"references-count":36,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2008,12,1]]}},"URL":"https:\/\/doi.org\/10.1214\/08-aos600","relation":{},"ISSN":["0090-5364","2168-8966"],"issn-type":[{"value":"0090-5364","type":"print"},{"value":"2168-8966","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,12,1]]}}}