Sparse covariance matrix estimation in high-dimensional deconvolution. (English) Zbl 1466.62332

Summary: We study the estimation of the covariance matrix \(\Sigma\) of a \(p\)-dimensional normal random vector based on \(n\) independent observations corrupted by additive noise. Only a general nonparametric assumption is imposed on the distribution of the noise without any sparsity constraint on its covariance matrix. In this high-dimensional semiparametric deconvolution problem, we propose spectral thresholding estimators that are adaptive to the sparsity of \(\Sigma\). We establish an oracle inequality for these estimators under model miss-specification and derive non-asymptotic minimax convergence rates that are shown to be logarithmic in \(n/\log p\). We also discuss the estimation of low-rank matrices based on indirect observations as well as the generalization to elliptical distributions. The finite sample performance of the threshold estimators is illustrated in a numerical example.


62H12 Estimation in multivariate analysis
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference


Full Text: DOI arXiv Euclid


[1] Belomestny, D. and Reiß, M. (2006). Spectral calibration of exponential Lévy models. Finance Stoch.10 449-474. · Zbl 1126.91022
[2] Belomestny, D. and Trabs, M. (2017). Low-rank diffusion matrix estimation for high-dimensional time-changed Lévy processes. Ann. Inst. Henri Poincaré Probab. Stat. To appear. Available at arXiv:1510.04638. · Zbl 1404.62081
[3] Belomestny, D., Trabs, M. and Tsybakov, A.B. (2019). Supplement to “Sparse covariance matrix estimation in high-dimensional deconvolution”: A bound for weighted \(L^2\)-distances of certain densities. DOI:10.3150/18-BEJ1040ASUPP. · Zbl 1466.62332
[4] Bickel, P.J. and Levina, E. (2008). Covariance regularization by thresholding. Ann. Statist.36 2577-2604. · Zbl 1196.62062
[5] Bickel, P.J. and Levina, E. (2008). Regularized estimation of large covariance matrices. Ann. Statist.36 199-227. · Zbl 1132.62040
[6] Butucea, C. and Matias, C. (2005). Minimax estimation of the noise level and of the deconvolution density in a semiparametric convolution model. Bernoulli11 309-340. · Zbl 1063.62044
[7] Butucea, C., Matias, C. and Pouet, C. (2008). Adaptivity in convolution models with partially known noise distribution. Electron. J. Stat.2 897-915. · Zbl 1320.62066
[8] Butucea, C. and Tsybakov, A.B. (2008). Sharp optimality in density deconvolution with dominating bias. I. Theory Probab. Appl.52 24-39. · Zbl 1141.62021
[9] Cai, T. and Liu, W. (2011). Adaptive thresholding for sparse covariance matrix estimation. J. Amer. Statist. Assoc.106 672-684. · Zbl 1232.62086
[10] Cai, T.T., Ren, Z. and Zhou, H.H. (2016). Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation. Electron. J. Stat.10 1-59. · Zbl 1331.62272
[11] Cai, T.T. and Zhang, A. (2016). Minimax rate-optimal estimation of high-dimensional covariance matrices with incomplete data. J. Multivariate Anal.150 55-74. · Zbl 1347.62088
[12] Cai, T.T., Zhang, C.-H. and Zhou, H.H. (2010). Optimal rates of convergence for covariance matrix estimation. Ann. Statist.38 2118-2144. · Zbl 1202.62073
[13] Cai, T.T. and Zhou, H.H. (2012). Minimax estimation of large covariance matrices under \(\ell_1\)-norm. Statist. Sinica22 1319-1349. · Zbl 1266.62036
[14] Carroll, R.J. and Hall, P. (1988). Optimal rates of convergence for deconvolving a density. J. Amer. Statist. Assoc.83 1184-1186. · Zbl 0673.62033
[15] Comte, F. and Lacour, C. (2011). Data-driven density estimation in the presence of additive noise with unknown distribution. J. R. Stat. Soc. Ser. B. Stat. Methodol.73 601-627. · Zbl 1226.62034
[16] Cressie, N. and Wikle, C.K. (2011). Statistics for Spatio-Temporal Data. Wiley Series in Probability and Statistics. Hoboken, NJ: Wiley. · Zbl 1273.62017
[17] Dattner, I., Reiß, M. and Trabs, M. (2016). Adaptive quantile estimation in deconvolution with unknown error distribution. Bernoulli22 143-192. · Zbl 1388.62095
[18] Delaigle, A. and Hall, P. (2016). Methodology for non-parametric deconvolution when the error distribution is unknown. J. R. Stat. Soc. Ser. B. Stat. Methodol.78 231-252. · Zbl 1411.62092
[19] Delaigle, A., Hall, P. and Meister, A. (2008). On deconvolution with repeated measurements. Ann. Statist.36 665-685. · Zbl 1133.62026
[20] Delaigle, A. and Meister, A. (2011). Nonparametric function estimation under Fourier-oscillating noise. Statist. Sinica21 1065-1092. · Zbl 1232.62057
[21] Eckle, K., Bissantz, N. and Dette, H. (2017). Multiscale inference for multivariate deconvolution. Electron. J. Stat.11 4179-4219. · Zbl 1380.62143
[22] El Karoui, N. (2008). Operator norm consistent estimation of large-dimensional sparse covariance matrices. Ann. Statist.36 2717-2756. · Zbl 1196.62064
[23] Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist.19 1257-1272. · Zbl 0729.62033
[24] Fan, J., Li, Y. and Yu, K. (2012). Vast volatility matrix estimation using high-frequency data for portfolio selection. J. Amer. Statist. Assoc.107 412-428. · Zbl 1328.91266
[25] Fan, J., Liao, Y. and Liu, H. (2016). An overview of the estimation of large covariance and precision matrices. Econom. J.19 C1-C32. · Zbl 1521.62083
[26] Fan, J., Liao, Y. and Mincheva, M. (2011). High-dimensional covariance matrix estimation in approximate factor models. Ann. Statist.39 3320-3356. · Zbl 1246.62151
[27] 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. · Zbl 1411.62138
[28] Fang, K.T., Kotz, S. and Ng, K.W. (1990). Symmetric Multivariate and Related Distributions. Monographs on Statistics and Applied Probability36. London: Chapman & Hall. · Zbl 0699.62048
[29] Friedman, J., Hastie, T. and Tibshirani, R. (2008). Sparse inverse covariance estimation with the graphical lasso. Biostatistics9 432-441. · Zbl 1143.62076
[30] Giné, E. and Nickl, R. (2016). Mathematical Foundations of Infinite-Dimensional Statistical Models. Cambridge Series in Statistical and Probabilistic Mathematics. New York: Cambridge Univ. Press. · Zbl 1358.62014
[31] Jacod, J. and Reiss, M. (2014). A remark on the rates of convergence for integrated volatility estimation in the presence of jumps. Ann. Statist.42 1131-1144. · Zbl 1305.62036
[32] Johannes, J. (2009). Deconvolution with unknown error distribution. Ann. Statist.37 2301-2323. · Zbl 1173.62018
[33] Kappus, J. and Mabon, G. (2014). Adaptive density estimation in deconvolution problems with unknown error distribution. Electron. J. Stat.8 2879-2904. · Zbl 1308.62074
[34] Koltchinskii, V., Lounici, K. and Tsybakov, A.B. (2011). Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion. Ann. Statist.39 2302-2329. · Zbl 1231.62097
[35] Lam, C. and Fan, J. (2009). Sparsistency and rates of convergence in large covariance matrix estimation. Ann. Statist.37 4254-4278. · Zbl 1191.62101
[36] Lepski, O. and Willer, T. (2017). Estimation in the convolution structure density model. Part I: Oracle inequalities. Preprint. Available at arXiv:1704.04418. · Zbl 1380.62208
[37] Lepski, O. and Willer, T. (2017). Estimation in the convolution structure density model. Part II: Adaptation over the scale of anisotropic classes. Preprint. Available at arXiv:1704.04420. · Zbl 1380.62208
[38] Lounici, K. (2014). High-dimensional covariance matrix estimation with missing observations. Bernoulli20 1029-1058. · Zbl 1320.62124
[39] Low, M.G. (1997). On nonparametric confidence intervals. Ann. Statist.25 2547-2554. · Zbl 0894.62055
[40] Masry, E. (1993). Strong consistency and rates for deconvolution of multivariate densities of stationary processes. Stochastic Process. Appl.47 53-74. · Zbl 0797.62071
[41] Matias, C. (2002). Semiparametric deconvolution with unknown noise variance. ESAIM Probab. Stat.6 271-292.
[42] Meister, A. (2008). Deconvolution from Fourier-oscillating error densities under decay and smoothness restrictions. Inverse Probl.24 015003, 14. · Zbl 1143.65106
[43] Neumann, M.H. (1997). On the effect of estimating the error density in nonparametric deconvolution. J. Nonparametr. Stat.7 307-330. · Zbl 1003.62514
[44] Rigollet, P. and Tsybakov, A. (2011). Exponential screening and optimal rates of sparse estimation. Ann. Statist.39 731-771. · Zbl 1215.62043
[45] Rigollet, P. and Tsybakov, A.B. (2012). Comment: “Minimax estimation of large covariance matrices under \(\ell_1\)-norm” [MR3027084]. Statist. Sinica22 1358-1367. · Zbl 1295.62057
[46] Rothman, A.J. (2012). Positive definite estimators of large covariance matrices. Biometrika99 733-740. · Zbl 1437.62595
[47] Rothman, A.J., Levina, E. and Zhu, J. (2009). Generalized thresholding of large covariance matrices. J. Amer. Statist. Assoc.104 177-186. · Zbl 1388.62170
[48] Sanandaji, B.M., Tascikaraoglu, A., Poolla, K. and Varaiya, P. (2015). Low-dimensional models in spatio-temporal wind speed forecasting. In American Control Conference (ACC) 4485-4490. IEEE.
[49] Sato, K. (2013). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics68. Cambridge: Cambridge Univ. Press. Translated from the 1990 Japanese original, Revised edition of the 1999 English translation.
[50] Tao, M., Wang, Y. and Zhou, H.H. (2013). Optimal sparse volatility matrix estimation for high-dimensional Itô processes with measurement errors. Ann. Statist.41 1816-1864. · Zbl 1281.62178
[51] Tsybakov, A.B. (2009). Introduction to Nonparametric Estimation. Springer Series in Statistics. New York: Springer. Revised and extended from the 2004 French original, Translated by Vladimir Zaiats. · Zbl 1176.62032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.