## Bernoulli

• Bernoulli
• Volume 25, Number 3 (2019), 1901-1938.

### Sparse covariance matrix estimation in high-dimensional deconvolution

#### Abstract

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.

#### Article information

Source
Bernoulli, Volume 25, Number 3 (2019), 1901-1938.

Dates
Revised: March 2018
First available in Project Euclid: 12 June 2019

https://projecteuclid.org/euclid.bj/1560326432

Digital Object Identifier
doi:10.3150/18-BEJ1040A

Mathematical Reviews number (MathSciNet)
MR3961235

Zentralblatt MATH identifier
07066244

#### Citation

Belomestny, Denis; Trabs, Mathias; Tsybakov, Alexandre B. Sparse covariance matrix estimation in high-dimensional deconvolution. Bernoulli 25 (2019), no. 3, 1901--1938. doi:10.3150/18-BEJ1040A. https://projecteuclid.org/euclid.bj/1560326432

#### References

• [1] Belomestny, D. and Reiß, M. (2006). Spectral calibration of exponential Lévy models. Finance Stoch. 10 449–474.
• [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.
• [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.
• [4] Bickel, P.J. and Levina, E. (2008). Covariance regularization by thresholding. Ann. Statist. 36 2577–2604.
• [5] Bickel, P.J. and Levina, E. (2008). Regularized estimation of large covariance matrices. Ann. Statist. 36 199–227.
• [6] Butucea, C. and Matias, C. (2005). Minimax estimation of the noise level and of the deconvolution density in a semiparametric convolution model. Bernoulli 11 309–340.
• [7] Butucea, C., Matias, C. and Pouet, C. (2008). Adaptivity in convolution models with partially known noise distribution. Electron. J. Stat. 2 897–915.
• [8] Butucea, C. and Tsybakov, A.B. (2008). Sharp optimality in density deconvolution with dominating bias. I. Theory Probab. Appl. 52 24–39.
• [9] Cai, T. and Liu, W. (2011). Adaptive thresholding for sparse covariance matrix estimation. J. Amer. Statist. Assoc. 106 672–684.
• [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.
• [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.
• [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.
• [13] Cai, T.T. and Zhou, H.H. (2012). Minimax estimation of large covariance matrices under $\ell_{1}$-norm. Statist. Sinica 22 1319–1349.
• [14] Carroll, R.J. and Hall, P. (1988). Optimal rates of convergence for deconvolving a density. J. Amer. Statist. Assoc. 83 1184–1186.
• [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.
• [16] Cressie, N. and Wikle, C.K. (2011). Statistics for Spatio-Temporal Data. Wiley Series in Probability and Statistics. Hoboken, NJ: Wiley.
• [17] Dattner, I., Reiß, M. and Trabs, M. (2016). Adaptive quantile estimation in deconvolution with unknown error distribution. Bernoulli 22 143–192.
• [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.
• [19] Delaigle, A., Hall, P. and Meister, A. (2008). On deconvolution with repeated measurements. Ann. Statist. 36 665–685.
• [20] Delaigle, A. and Meister, A. (2011). Nonparametric function estimation under Fourier-oscillating noise. Statist. Sinica 21 1065–1092.
• [21] Eckle, K., Bissantz, N. and Dette, H. (2017). Multiscale inference for multivariate deconvolution. Electron. J. Stat. 11 4179–4219.
• [22] El Karoui, N. (2008). Operator norm consistent estimation of large-dimensional sparse covariance matrices. Ann. Statist. 36 2717–2756.
• [23] Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 1257–1272.
• [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.
• [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.
• [26] Fan, J., Liao, Y. and Mincheva, M. (2011). High-dimensional covariance matrix estimation in approximate factor models. Ann. Statist. 39 3320–3356.
• [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.
• [28] Fang, K.T., Kotz, S. and Ng, K.W. (1990). Symmetric Multivariate and Related Distributions. Monographs on Statistics and Applied Probability 36. London: Chapman & Hall.
• [29] Friedman, J., Hastie, T. and Tibshirani, R. (2008). Sparse inverse covariance estimation with the graphical lasso. Biostatistics 9 432–441.
• [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.
• [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.
• [32] Johannes, J. (2009). Deconvolution with unknown error distribution. Ann. Statist. 37 2301–2323.
• [33] Kappus, J. and Mabon, G. (2014). Adaptive density estimation in deconvolution problems with unknown error distribution. Electron. J. Stat. 8 2879–2904.
• [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.
• [35] Lam, C. and Fan, J. (2009). Sparsistency and rates of convergence in large covariance matrix estimation. Ann. Statist. 37 4254–4278.
• [36] Lepski, O. and Willer, T. (2017). Estimation in the convolution structure density model. Part I: Oracle inequalities. Preprint. Available at arXiv:1704.04418.
• [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.
• [38] Lounici, K. (2014). High-dimensional covariance matrix estimation with missing observations. Bernoulli 20 1029–1058.
• [39] Low, M.G. (1997). On nonparametric confidence intervals. Ann. Statist. 25 2547–2554.
• [40] Masry, E. (1993). Strong consistency and rates for deconvolution of multivariate densities of stationary processes. Stochastic Process. Appl. 47 53–74.
• [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.
• [43] Neumann, M.H. (1997). On the effect of estimating the error density in nonparametric deconvolution. J. Nonparametr. Stat. 7 307–330.
• [44] Rigollet, P. and Tsybakov, A. (2011). Exponential screening and optimal rates of sparse estimation. Ann. Statist. 39 731–771.
• [45] Rigollet, P. and Tsybakov, A.B. (2012). Comment: “Minimax estimation of large covariance matrices under $\ell_{1}$-norm” [MR3027084]. Statist. Sinica 22 1358–1367.
• [46] Rothman, A.J. (2012). Positive definite estimators of large covariance matrices. Biometrika 99 733–740.
• [47] Rothman, A.J., Levina, E. and Zhu, J. (2009). Generalized thresholding of large covariance matrices. J. Amer. Statist. Assoc. 104 177–186.
• [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 Mathematics 68. 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.
• [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.
• [52] Tsybakov, A.B. (2013). Aggregation and high-dimensional statistics. Saint Flour lecture notes.

#### Supplemental materials

• Supplement: A bound for weighted $L^{2}$-distances of certain densities. We prove Proposition 16 which is needed to show the lower bound for the covariance estimation in the deconvolution model. More precisely, a bound for the $L^{2}$-distance of the certain densities with polynomial weights is proved.