The Annals of Statistics

Large covariance estimation through elliptical factor models

Jianqing Fan, Han Liu, and Weichen Wang

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Abstract

We propose a general Principal Orthogonal complEment Thresholding (POET) framework for large-scale covariance matrix estimation based on the approximate factor model. A set of high-level sufficient conditions for the procedure to achieve optimal rates of convergence under different matrix norms is established to better understand how POET works. Such a framework allows us to recover existing results for sub-Gaussian data in a more transparent way that only depends on the concentration properties of the sample covariance matrix. As a new theoretical contribution, for the first time, such a framework allows us to exploit conditional sparsity covariance structure for the heavy-tailed data. In particular, for the elliptical distribution, we propose a robust estimator based on the marginal and spatial Kendall’s tau to satisfy these conditions. In addition, we study conditional graphical model under the same framework. The technical tools developed in this paper are of general interest to high-dimensional principal component analysis. Thorough numerical results are also provided to back up the developed theory.

Article information

Source
Ann. Statist., Volume 46, Number 4 (2018), 1383-1414.

Dates
Received: July 2015
Revised: January 2017
First available in Project Euclid: 27 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.aos/1530086420

Digital Object Identifier
doi:10.1214/17-AOS1588

Mathematical Reviews number (MathSciNet)
MR3819104

Zentralblatt MATH identifier
06936465

Subjects
Primary: 62H25: Factor analysis and principal components; correspondence analysis
Secondary: 62H12: Estimation

Keywords
Principal component analysis approximate factor model sub-Gaussian family elliptical distribution conditional graphical model marginal and spatial Kendall’s tau

Citation

Fan, Jianqing; Liu, Han; Wang, Weichen. Large covariance estimation through elliptical factor models. Ann. Statist. 46 (2018), no. 4, 1383--1414. doi:10.1214/17-AOS1588. https://projecteuclid.org/euclid.aos/1530086420


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References

  • Agarwal, A., Negahban, S. and Wainwright, M. J. (2012). Noisy matrix decomposition via convex relaxation: Optimal rates in high dimensions. Ann. Statist. 40 1171–1197.
    Mathematical Reviews (MathSciNet): MR2985947
    Zentralblatt MATH: 1274.62219
    Digital Object Identifier: doi:10.1214/12-AOS1000
    Project Euclid: euclid.aos/1342625465
  • Amini, A. A. and Wainwright, M. J. (2009). High-dimensional analysis of semidefinite relaxations for sparse principal components. Ann. Statist. 37 2877–2921.
    Zentralblatt MATH: 1173.62049
    Digital Object Identifier: doi:10.1214/08-AOS664
    Project Euclid: euclid.aos/1247836672
  • Antoniadis, A. and Fan, J. (2001). Regularization of wavelet approximations. J. Amer. Statist. Assoc. 96 939–967.
    Mathematical Reviews (MathSciNet): MR1946364
    Zentralblatt MATH: 1072.62561
    Digital Object Identifier: doi:10.1198/016214501753208942
  • Bai, J. and Li, K. (2012). Statistical analysis of factor models of high dimension. Ann. Statist. 40 436–465.
    Mathematical Reviews (MathSciNet): MR3014313
    Zentralblatt MATH: 1246.62144
    Digital Object Identifier: doi:10.1214/11-AOS966
    Project Euclid: euclid.aos/1334581749
  • Bai, J. and Ng, S. (2002). Determining the number of factors in approximate factor models. Econometrica 70 191–221.
    Zentralblatt MATH: 1103.91399
    Digital Object Identifier: doi:10.1111/1468-0262.00273
  • Bai, J. and Ng, S. (2013). Principal components estimation and identification of static factors. J. Econometrics 176 18–29.
    Zentralblatt MATH: 1284.62350
    Digital Object Identifier: doi:10.1016/j.jeconom.2013.03.007
  • Belloni, A. and Chernozhukov, V. (2011). $\ell_{1}$-penalized quantile regression in high-dimensional sparse models. Ann. Statist. 39 82–130.
    Zentralblatt MATH: 1209.62064
    Digital Object Identifier: doi:10.1214/10-AOS827
    Project Euclid: euclid.aos/1291388370
  • Beran, R. (1978). An efficient and robust adaptive estimator of location. Ann. Statist. 6 292–313.
    Mathematical Reviews (MathSciNet): MR0518885
    Zentralblatt MATH: 0378.62051
    Digital Object Identifier: doi:10.1214/aos/1176344125
    Project Euclid: euclid.aos/1176344125
  • Berthet, Q. and Rigollet, P. (2013a). Optimal detection of sparse principal components in high dimension. Ann. Statist. 41 1780–1815.
    Zentralblatt MATH: 1277.62155
    Digital Object Identifier: doi:10.1214/13-AOS1127
    Project Euclid: euclid.aos/1378386239
  • Berthet, Q. and Rigollet, P. (2013b). Complexity theoretic lower bounds for sparse principal component detection. In Proceedings of the 26th Annual Conference on Learning Theory 1046–1066. PMLR, Princeton, NJ.
    Zentralblatt MATH: 1277.62155
    Digital Object Identifier: doi:10.1214/13-AOS1127
    Project Euclid: euclid.aos/1378386239
  • Bickel, P. J. (1982). On adaptive estimation. Ann. Statist. 10 647–671.
    Zentralblatt MATH: 0489.62033
    Digital Object Identifier: doi:10.1214/aos/1176345863
    Project Euclid: euclid.aos/1176345863
  • Bickel, P. J. and Levina, E. (2008a). Covariance regularization by thresholding. Ann. Statist. 36 2577–2604.
    Zentralblatt MATH: 1196.62062
    Digital Object Identifier: doi:10.1214/08-AOS600
    Project Euclid: euclid.aos/1231165180
  • Bickel, P. J. and Levina, E. (2008b). Regularized estimation of large covariance matrices. Ann. Statist. 36 199–227.
    Zentralblatt MATH: 1132.62040
    Digital Object Identifier: doi:10.1214/009053607000000758
    Project Euclid: euclid.aos/1201877299
  • Birnbaum, A., Johnstone, I. M., Nadler, B. and Paul, D. (2013). Minimax bounds for sparse PCA with noisy high-dimensional data. Ann. Statist. 41 1055–1084.
    Zentralblatt MATH: 1292.62071
    Digital Object Identifier: doi:10.1214/12-AOS1014
    Project Euclid: euclid.aos/1371150893
  • Cai, T. and Liu, W. (2011). Adaptive thresholding for sparse covariance matrix estimation. J. Amer. Statist. Assoc. 106 672–684.
    Zentralblatt MATH: 1232.62086
    Digital Object Identifier: doi:10.1198/jasa.2011.tm10560
  • Cai, T., Liu, W. and Luo, X. (2011). A constrained $\ell_{1}$ minimization approach to sparse precision matrix estimation. J. Amer. Statist. Assoc. 106 594–607.
  • Cai, T. T., Ma, Z. and Wu, Y. (2013). Sparse PCA: Optimal rates and adaptive estimation. Ann. Statist. 41 3074–3110.
    Zentralblatt MATH: 1288.62099
    Digital Object Identifier: doi:10.1214/13-AOS1178
    Project Euclid: euclid.aos/1388545679
  • Cai, T., Ma, Z. and Wu, Y. (2015). Optimal estimation and rank detection for sparse spiked covariance matrices. Probab. Theory Related Fields 161 781–815.
    Mathematical Reviews (MathSciNet): MR3334281
    Zentralblatt MATH: 1314.62130
    Digital Object Identifier: doi:10.1007/s00440-014-0562-z
  • Cai, T. T., Ren, Z. and Zhou, H. H. (2013). Optimal rates of convergence for estimating Toeplitz covariance matrices. Probab. Theory Related Fields 156 101–143.
    Zentralblatt MATH: 06176807
    Digital Object Identifier: doi:10.1007/s00440-012-0422-7
  • Cai, T. T., Zhang, C.-H. and Zhou, H. H. (2010). Optimal rates of convergence for covariance matrix estimation. Ann. Statist. 38 2118–2144.
    Mathematical Reviews (MathSciNet): MR2676885
    Zentralblatt MATH: 1202.62073
    Digital Object Identifier: doi:10.1214/09-AOS752
    Project Euclid: euclid.aos/1278861244
  • Cai, T. T. and Zhou, H. H. (2012). Optimal rates of convergence for sparse covariance matrix estimation. Ann. Statist. 40 2389–2420.
    Zentralblatt MATH: 1373.62247
    Digital Object Identifier: doi:10.1214/12-AOS998
    Project Euclid: euclid.aos/1359987525
  • Cai, T. T., Li, H., Liu, W. and Xie, J. (2013). Covariate-adjusted precision matrix estimation with an application in genetical genomics. Biometrika 100 139–156.
    Zentralblatt MATH: 1284.62648
    Digital Object Identifier: doi:10.1093/biomet/ass058
  • Catoni, O. (2012). Challenging the empirical mean and empirical variance: A deviation study. Ann. Inst. Henri Poincaré Probab. Stat. 48 1148–1185.
    Zentralblatt MATH: 1282.62070
    Digital Object Identifier: doi:10.1214/11-AIHP454
    Project Euclid: euclid.aihp/1353098444
  • Chamberlain, G. and Rothschild, M. (1983). Arbitrage, factor structure, and mean-variance analysis on large asset markets. Econometrica 51 1281–1304.
    Zentralblatt MATH: 0523.90017
    Digital Object Identifier: doi:10.2307/1912275
  • Chandrasekaran, V., Parrilo, P. A. and Willsky, A. S. (2012). Latent variable graphical model selection via convex optimization. Ann. Statist. 40 1935–1967.
    Zentralblatt MATH: 1257.62061
    Digital Object Identifier: doi:10.1214/11-AOS949
    Project Euclid: euclid.aos/1351602527
  • Chandrasekaran, V., Sanghavi, S., Parrilo, P. A. and Willsky, A. S. (2011). Rank-sparsity incoherence for matrix decomposition. SIAM J. Optim. 21 572–596.
    Zentralblatt MATH: 1226.90067
    Digital Object Identifier: doi:10.1137/090761793
  • Choi, K. and Marden, J. (1998). A multivariate version of Kendall’s $\tau$. J. Nonparametr. Stat. 9 261–293.
  • Christensen, D. (2005). Fast algorithms for the calculation of Kendall’s $\tau$. Comput. Statist. 20 51–62.
  • Čížek, P., Härdle, W. and Weron, R., eds. (2005). Statistical Tools for Finance and Insurance. Springer, Berlin.
  • Croux, C., Ollila, E. and Oja, H. (2002). Sign and rank covariance matrices: Statistical properties and application to principal components analysis. In Statistical Data Analysis Based on the $L_{1}$-Norm and Related Methods (Neuchâtel, 2002) 257–269. Birkhäuser, Basel.
    Zentralblatt MATH: 1145.62343
  • Davis, C. and Kahan, W. M. (1970). The rotation of eigenvectors by a perturbation. III. SIAM J. Numer. Anal. 7 1–46.
    Zentralblatt MATH: 0198.47201
    Digital Object Identifier: doi:10.1137/0707001
  • Dupuis Lozeron, E. and Victoria-Feser, M. P. (2010). Robust estimation of constrained covariance matrices for confirmatory factor analysis. Comput. Statist. Data Anal. 54 3020–3032.
    Mathematical Reviews (MathSciNet): MR2727731
    Zentralblatt MATH: 1284.62196
    Digital Object Identifier: doi:10.1016/j.csda.2009.08.014
  • Dürre, A., Vogel, D. and Tyler, D. E. (2014). The spatial sign covariance matrix with unknown location. J. Multivariate Anal. 130 107–117.
    Zentralblatt MATH: 1292.62079
    Digital Object Identifier: doi:10.1016/j.jmva.2014.05.004
  • El Karoui, N. (2008). Operator norm consistent estimation of large-dimensional sparse covariance matrices. Ann. Statist. 36 2717–2756.
    Zentralblatt MATH: 1196.62064
    Digital Object Identifier: doi:10.1214/07-AOS559
    Project Euclid: euclid.aos/1231165183
  • Fan, J., Fan, Y. and Barut, E. (2014). Adaptive robust variable selection. Ann. Statist. 42 324–351.
    Zentralblatt MATH: 1296.62144
    Digital Object Identifier: doi:10.1214/13-AOS1191
    Project Euclid: euclid.aos/1395234980
  • Fan, J., Fan, Y. and Lv, J. (2008). High dimensional covariance matrix estimation using a factor model. J. Econometrics 147 186–197.
    Zentralblatt MATH: 06596553
    Digital Object Identifier: doi:10.1016/j.jeconom.2008.09.017
  • Fan, J., Feng, Y. and Wu, Y. (2009). Network exploration via the adaptive lasso and SCAD penalties. Ann. Appl. Stat. 3 521–541.
    Zentralblatt MATH: 1166.62040
    Digital Object Identifier: doi:10.1214/08-AOAS215
    Project Euclid: euclid.aoas/1245676184
  • Fan, J., Li, Q. and Wang, Y. (2017). Estimation of high dimensional mean regression in the absence of symmetry and light tail assumptions. J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 247–265.
  • Fan, J., Liao, Y. and Mincheva, M. (2011). High-dimensional covariance matrix estimation in approximate factor models. Ann. Statist. 39 3320–3356.
    Zentralblatt MATH: 1246.62151
    Digital Object Identifier: doi:10.1214/11-AOS944
    Project Euclid: euclid.aos/1330958681
  • Fan, J., Liao, Y. and Mincheva, M. (2013). Large covariance estimation by thresholding principal orthogonal complements. J. R. Stat. Soc. Ser. B. Stat. Methodol. 75 603–680. With 33 discussions by 57 authors and a reply by Fan, Liao and Mincheva.
  • Fan, J., Liao, Y. and Wang, W. (2016). Projected principal component analysis in factor models. Ann. Statist. 44 219–254.
    Zentralblatt MATH: 1331.62295
    Digital Object Identifier: doi:10.1214/15-AOS1364
    Project Euclid: euclid.aos/1449755962
  • Fan, J., Liu, H. and Wang, W. (2018). Supplement to “Large covariance estimation through elliptical factor models”. DOI:10.1214/17-AOS1588SUPP.
  • Fan, J., Wang, W. and Zhong, Y. (2016). Robust covariance estimation for approximate factor models. Available at arXiv:1602.00719.
    arXiv: 1602.00719
  • Fan, J., Wang, W. and Zhong, Y. (2017). An $\ell_{\infty}$ eigenvector perturbation bound and its application to robust covariance estimation. Available at arXiv:1603.03516.
    arXiv: 1603.03516
  • Fan, J., Ke, Z. T., Liu, H. and Xia, L. (2015). QUADRO: A supervised dimension reduction method via Rayleigh quotient optimization. Ann. Statist. 43 1498–1534.
    Zentralblatt MATH: 1317.62054
    Digital Object Identifier: doi:10.1214/14-AOS1307
    Project Euclid: euclid.aos/1434546213
  • Fan, J., Liu, H., Wang, W. and Zhu, Z. (2016). Heterogeneity adjustment with applications to graphical model inference. Available at arXiv:1602.05455.
    arXiv: 1602.05455
  • Fang, K. T., Kotz, S. and Ng, K. W. (1990). Symmetric Multivariate and Related Distributions. Monographs on Statistics and Applied Probability 36. Chapman & Hall, London.
    Zentralblatt MATH: 0699.62048
  • Friedman, J., Hastie, T. and Tibshirani, R. (2008). Sparse inverse covariance estimation with the graphical lasso. Biostatistics 9 432–441.
    Zentralblatt MATH: 1143.62076
    Digital Object Identifier: doi:10.1093/biostatistics/kxm045
  • Hallin, M. and Paindaveine, D. (2006). Semiparametrically efficient rank-based inference for shape. I. Optimal rank-based tests for sphericity. Ann. Statist. 34 2707–2756.
    Zentralblatt MATH: 1114.62066
    Digital Object Identifier: doi:10.1214/009053606000000731
    Project Euclid: euclid.aos/1179935063
  • Hampel, F. R. (1974). The influence curve and its role in robust estimation. J. Amer. Statist. Assoc. 69 383–393.
    Zentralblatt MATH: 0305.62031
    Digital Object Identifier: doi:10.1080/01621459.1974.10482962
  • Han, F. and Liu, H. (2014). Scale-invariant sparse PCA on high-dimensional meta-elliptical data. J. Amer. Statist. Assoc. 109 275–287.
    Zentralblatt MATH: 1367.62185
    Digital Object Identifier: doi:10.1080/01621459.2013.844699
  • Han, F. and Liu, H. (2018). ECA: High dimensional elliptical component analysis in non-Gaussian distributions. J. Amer. Statist. Assoc. 113 252–268.
  • Han, F. and Liu, H. (2017). Statistical analysis of latent generalized correlation matrix estimation in transelliptical distribution. Bernoulli 23 23–57.
    Mathematical Reviews (MathSciNet): MR3556765
    Zentralblatt MATH: 1359.62186
    Digital Object Identifier: doi:10.3150/15-BEJ702
    Project Euclid: euclid.bj/1475001347
  • Han, F., Lu, J. and Liu, H. (2014). Robust scatter matrix estimation for high dimensional distributions with heavy tails. Technical report.
  • Hsu, D., Kakade, S. M. and Zhang, T. (2011). Robust matrix decomposition with sparse corruptions. IEEE Trans. Inform. Theory 57 7221–7234.
    Zentralblatt MATH: 1365.15018
    Digital Object Identifier: doi:10.1109/TIT.2011.2158250
  • Hsu, D. and Sabato, S. (2014). Heavy-tailed regression with a generalized median-of-means. In Proceedings of the 31st International Conference on Machine Learning (ICML-14) 37–45. PMLR, Bejing, China.
  • Huber, P. J. (1964). Robust estimation of a location parameter. Ann. Math. Stat. 35 73–101.
    Zentralblatt MATH: 0136.39805
    Digital Object Identifier: doi:10.1214/aoms/1177703732
    Project Euclid: euclid.aoms/1177703732
  • Hult, H. and Lindskog, F. (2002). Multivariate extremes, aggregation and dependence in elliptical distributions. Adv. in Appl. Probab. 34 587–608.
    Zentralblatt MATH: 1023.60021
    Digital Object Identifier: doi:10.1239/aap/1033662167
    Project Euclid: euclid.aap/1033662167
  • 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.
    Zentralblatt MATH: 06441085
    Digital Object Identifier: doi:10.1198/jasa.2009.0121
  • Kendall, M. G. (1948). Rank Correlation Methods. C. Griffin, London.
    Zentralblatt MATH: 0032.17602
  • Knight, W. R. (1966). A computer method for calculating Kendall’s tau with ungrouped data. J. Amer. Statist. Assoc. 61 436–439.
    Zentralblatt MATH: 0138.13202
    Digital Object Identifier: doi:10.1080/01621459.1966.10480879
  • Koenker, R. (2005). Quantile Regression. Econometric Society Monographs 38. Cambridge Univ. Press Cambridge.
  • Lam, C. and Fan, J. (2009). Sparsistency and rates of convergence in large covariance matrix estimation. Ann. Statist. 37 4254–4278.
    Zentralblatt MATH: 1191.62101
    Digital Object Identifier: doi:10.1214/09-AOS720
    Project Euclid: euclid.aos/1256303543
  • Levina, E. and Vershynin, R. (2012). Partial estimation of covariance matrices. Probab. Theory Related Fields 153 405–419.
    Zentralblatt MATH: 1318.62179
    Digital Object Identifier: doi:10.1007/s00440-011-0349-4
  • Liu, H., Han, F. and Zhang, C. (2012). Transelliptical graphical models. In Advances in Neural Information Processing Systems 800–808. Curran Associates, Inc., Lake Tahoe, NV.
  • Liu, L., Hawkins, D. M., Ghosh, S. and Young, S. S. (2003). Robust singular value decomposition analysis of microarray data. Proc. Natl. Acad. Sci. USA 100 13167–13172.
    Zentralblatt MATH: 1108.62117
    Digital Object Identifier: doi:10.1073/pnas.1733249100
  • Ma, Z. (2013). Sparse principal component analysis and iterative thresholding. Ann. Statist. 41 772–801.
    Zentralblatt MATH: 1267.62074
    Digital Object Identifier: doi:10.1214/13-AOS1097
    Project Euclid: euclid.aos/1368018173
  • Marden, J. I. (1999). Some robust estimates of principal components. Statist. Probab. Lett. 43 349–359.
    Zentralblatt MATH: 0939.62055
    Digital Object Identifier: doi:10.1016/S0167-7152(98)00272-7
  • Meinshausen, N. and Bühlmann, P. (2006). High-dimensional graphs and variable selection with the lasso. Ann. Statist. 34 1436–1462.
    Zentralblatt MATH: 1113.62082
    Digital Object Identifier: doi:10.1214/009053606000000281
    Project Euclid: euclid.aos/1152540754
  • Mitra, R. and Zhang, C.-H. (2014). Multivariate analysis of nonparametric estimates of large correlation matrices. Available at arXiv:1403.6195.
    arXiv: 1403.6195
  • Pang, H., Liu, H. and Vanderbei, R. (2014). The fastclime package for linear programming and large-scale precision matrix estimation in R. J. Mach. Learn. Res. 15 489–493.
    Zentralblatt MATH: 1318.90002
  • Paul, D. and Johnstone, I. M. (2012). Augmented sparse principal component analysis for high dimensional data. Available at arXiv:1202.1242.
    arXiv: 1202.1242
  • Pison, G., Rousseeuw, P. J., Filzmoser, P. and Croux, C. (2003). Robust factor analysis. J. Multivariate Anal. 84 145–172.
    Zentralblatt MATH: 1038.62055
    Digital Object Identifier: doi:10.1016/S0047-259X(02)00007-6
  • Posekany, A., Felsenstein, K. and Sykacek, P. (2011). Biological assessment of robust noise models in microarray data analysis. Bioinformatics 27 807–814.
  • Rachev, S. T. (2003). Handbook of Heavy Tailed Distributions in Finance. Handbooks in Finance 1. Elsevier, Amsterdam.
  • Ravikumar, P., Wainwright, M. J., Raskutti, G. and Yu, B. (2011). High-dimensional covariance estimation by minimizing $\ell_{1}$-penalized log-determinant divergence. Electron. J. Stat. 5 935–980.
    Zentralblatt MATH: 1274.62190
    Digital Object Identifier: doi:10.1214/11-EJS631
  • Rothman, A. J., Levina, E. and Zhu, J. (2009). Generalized thresholding of large covariance matrices. J. Amer. Statist. Assoc. 104 177–186.
    Zentralblatt MATH: 06448242
    Digital Object Identifier: doi:10.1198/jasa.2009.0101
  • Rothman, A. J., Levina, E. and Zhu, J. (2010). Sparse multivariate regression with covariance estimation. J. Comput. Graph. Statist. 19 947–962. Supplementary materials available online.
  • Rousseeuw, P. J. and Croux, C. (1993). Alternatives to the median absolute deviation. J. Amer. Statist. Assoc. 88 1273–1283.
    Zentralblatt MATH: 0792.62025
    Digital Object Identifier: doi:10.1080/01621459.1993.10476408
  • Ruttimann, U. E., Unser, M., Rawlings, R. R., Rio, D., Ramsey, N. F., Mattay, V. S., Hommer, D. W., Frank, J. A. and Weinberger, D. R. (1998). Statistical analysis of functional MRI data in the wavelet domain. IEEE Trans. Med. Imag. 17 142–154.
  • Shen, D., Shen, H. and Marron, J. S. (2013). Consistency of sparse PCA in high dimension, low sample size contexts. J. Multivariate Anal. 115 317–333.
    Zentralblatt MATH: 1258.62072
    Digital Object Identifier: doi:10.1016/j.jmva.2012.10.007
  • Tsybakov, A. B. (2009). Introduction to Nonparametric Estimation. Springer, New York. Revised and extended from the 2004 French original. Translated by Vladimir Zaiats.
  • Tyler, D. E. (1982). Radial estimates and the test for sphericity. Biometrika 69 429–436.
    Zentralblatt MATH: 0501.62041
    Digital Object Identifier: doi:10.1093/biomet/69.2.429
  • Vanderbei, R. J. (2008). Linear Programming: Foundations and Extensions, 3rd ed. International Series in Operations Research & Management Science 114. Springer, New York.
  • Vershynin, R. (2012). Introduction to the non-asymptotic analysis of random matrices. In Compressed Sensing 210–268. Cambridge Univ. Press Cambridge.
  • Visuri, S., Koivunen, V. and Oja, H. (2000). Sign and rank covariance matrices. J. Statist. Plann. Inference 91 557–575.
    Zentralblatt MATH: 0965.62049
    Digital Object Identifier: doi:10.1016/S0378-3758(00)00199-3
  • Vogel, D. and Fried, R. (2011). Elliptical graphical modelling. Biometrika 98 935–951.
    Zentralblatt MATH: 1228.62069
    Digital Object Identifier: doi:10.1093/biomet/asr037
  • Vu, V. Q. and Lei, J. (2013). Minimax sparse principal subspace estimation in high dimensions. Ann. Statist. 41 2905–2947.
    Zentralblatt MATH: 1288.62103
    Digital Object Identifier: doi:10.1214/13-AOS1151
    Project Euclid: euclid.aos/1388545673
  • Wang, W. and Fan, J. (2017). Asymptotics of empirical eigenstructure for high dimensional spiked covariance. Ann. Statist. 45 1342–1374.
    Mathematical Reviews (MathSciNet): MR3662457
    Zentralblatt MATH: 1373.62299
    Digital Object Identifier: doi:10.1214/16-AOS1487
    Project Euclid: euclid.aos/1497319697
  • Wegkamp, M. and Zhao, Y. (2016). Adaptive estimation of the copula correlation matrix for semiparametric elliptical copulas. Bernoulli 22 1184–1226.
    Zentralblatt MATH: 06562309
    Digital Object Identifier: doi:10.3150/14-BEJ690
    Project Euclid: euclid.bj/1447077773
  • Wu, Y. and Liu, Y. (2009). Variable selection in quantile regression. Statist. Sinica 19 801–817.
    Zentralblatt MATH: 1166.62012
  • Yuan, M. (2010). High dimensional inverse covariance matrix estimation via linear programming. J. Mach. Learn. Res. 11 2261–2286.
    Zentralblatt MATH: 1242.62043
  • Yuan, M. and Lin, Y. (2007). Model selection and estimation in the Gaussian graphical model. Biometrika 94 19–35.
    Zentralblatt MATH: 1142.62408
    Digital Object Identifier: doi:10.1093/biomet/asm018
  • Zhao, T., Roeder, K. and Liu, H. (2014). Positive semidefinite rank-based correlation matrix estimation with application to semiparametric graph estimation. J. Comput. Graph. Statist. 23 895–922.
  • Zou, H., Hastie, T. and Tibshirani, R. (2006). Sparse principal component analysis. J. Comput. Graph. Statist. 15 265–286.
  • Zou, H. and Yuan, M. (2008). Composite quantile regression and the oracle model selection theory. Ann. Statist. 36 1108–1126.
    Zentralblatt MATH: 1360.62394
    Digital Object Identifier: doi:10.1214/07-AOS507
    Project Euclid: euclid.aos/1211819558

Supplemental materials

  • Technical proofs. This supplementary material contains all the remaining proofs and technical lemmas and the comparison of relative error norms.
    Digital Object Identifier: doi:10.1214/17-AOS1588SUPP
    Supplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.

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The Annals of Statistics

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