Thresholding and graphical local Whittle estimation

This work develops non-asymptotic theory for estimation of the long-run variance matrix and its inverse, the so-called precision matrix, for high-dimensional Gaussian time series under general assumptions on the dependence structure including long-range dependence. The estimation involves shrinkage techniques which are thresholding and penalizing versions of the classical multivariate local Whittle estimator. The results ensure consistent estimation in a double asymptotic regime where the number of component time series is allowed to grow with the sample size as long as the true model parameters are sparse. The key technical result is a concentration inequality of the local Whittle estimator for the long-run variance matrix around the true model parameters. In particular, it handles simultaneously the estimation of the memory parameters which enter the underlying model. Novel algorithms for the considered procedures are proposed, and a simulation study and a data application are also provided.

[1]  T. Cai,et al.  A Constrained ℓ1 Minimization Approach to Sparse Precision Matrix Estimation , 2011, 1102.2233.

[2]  Zehua Chen,et al.  EXTENDED BIC FOR SMALL-n-LARGE-P SPARSE GLM , 2012 .

[3]  Martin J. Wainwright,et al.  High-Dimensional Statistics , 2019 .

[4]  Minghua Lin,et al.  On a decomposition lemma for positive semi-definite block-matrices , 2012, 1202.0473.

[5]  V. Pipiras,et al.  DEFINITIONS AND REPRESENTATIONS OF MULTIVARIATE LONG‐RANGE DEPENDENT TIME SERIES , 2015 .

[6]  Jianqing Fan,et al.  Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties , 2001 .

[7]  É. Moulines,et al.  Log-Periodogram Regression Of Time Series With Long Range Dependence , 1999 .

[8]  Rina Foygel,et al.  Extended Bayesian Information Criteria for Gaussian Graphical Models , 2010, NIPS.

[9]  A. U.S.,et al.  Sparse Estimation of a Covariance Matrix , 2010 .

[10]  Katsumi Shimotsu,et al.  Gaussian semiparametric estimation of multivariate fractionally integrated processes , 2007 .

[11]  Vladas Pipiras,et al.  Asymptotic results for multivariate local Whittle estimation with applications , 2019, 2019 IEEE 8th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP).

[12]  P. Robinson.  Multiple Local Whittle Estimation in Stationary Systems , 2007, 0811.0948.

[13]  A. McD. Mercer.  Some new inequalities involving elementary mean values , 1999 .

[14]  Roman Vershynin,et al.  Introduction to the non-asymptotic analysis of random matrices , 2010, Compressed Sensing.

[15]  Jianqing Fan,et al.  An Overview of the Estimation of Large Covariance and Precision Matrices , 2015, The Econometrics Journal.

[16]  Vladas Pipiras,et al.  Long-Range Dependence and Self-Similarity , 2017 .

[17]  Harrison H. Zhou,et al.  Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation , 2016 .

[18]  Sumanta Basu,et al.  Large Spectral Density Matrix Estimation by Thresholding , 2018, 1812.00532.

[19]  P. Bickel,et al.  Regularized estimation of large covariance matrices , 2008, 0803.1909.

[20]  Jan Beran,et al.  Long-Memory Processes: Probabilistic Properties and Statistical Methods , 2013 .

[21]  Albrecht Böttcher,et al.  Norms of Toeplitz Matrices with Fisher-Hartwig Symbols , 2007, SIAM J. Matrix Anal. Appl..

[22]  Adam J. Rothman,et al.  Generalized Thresholding of Large Covariance Matrices , 2009 .

[23]  Yuehua Wu,et al.  TUNING PARAMETER SELECTION FOR PENALIZED LIKELIHOOD ESTIMATION OF GAUSSIAN GRAPHICAL MODEL , 2012 .

[24]  Chenlei Leng,et al.  Spectral analysis of high-dimensional time series , 2018, Electronic Journal of Statistics.

[25]  V. Pipiras,et al.  On distinguishing multiple changes in mean and long-range dependence using local Whittle estimation , 2014 .

[26]  Weidong Liu,et al.  Adaptive Thresholding for Sparse Covariance Matrix Estimation , 2011, 1102.2237.

[27]  Jiahua Chen,et al.  Extended Bayesian information criteria for model selection with large model spaces , 2008 .

[28]  V. Pipiras,et al.  Asymptotics of bivariate local Whittle estimators with applications to fractal connectivity , 2020 .

[29]  R. Dahlhaus.  Graphical interaction models for multivariate time series1 , 2000 .

[30]  Adam J. Rothman,et al.  Sparse permutation invariant covariance estimation , 2008, 0801.4837.

[31]  A. Böttcher,et al.  Weighted Markov‐type inequalities, norms of Volterra operators, and zeros of Bessel functions , 2010 .

[32]  R. Tibshirani,et al.  Sparse inverse covariance estimation with the graphical lasso. , 2008, Biostatistics.

[33]  K. Loparo,et al.  Inequalities for the trace of matrix product , 1994, IEEE Trans. Autom. Control..

[34]  Frank Nielsen.  Local Whittle estimation of multi‐variate fractionally integrated processes , 2011 .

[35]  P. Bickel,et al.  Covariance regularization by thresholding , 2009, 0901.3079.

[36]  Changryong Baek,et al.  Semiparametric, parametric, and possibly sparse models for multivariate long-range dependence , 2017, Optical Engineering + Applications.

[37]  M. Drton,et al.  Bayesian model choice and information criteria in sparse generalized linear models , 2011, 1112.5635.

[38]  Shiqian Ma,et al.  Sparse Inverse Covariance Selection via Alternating Linearization Methods , 2010, NIPS.

[39]  B. Nan,et al.  Estimation of large covariance and precision matrices from temporally dependent observations , 2014, The Annals of Statistics.

[40]  G. Michailidis,et al.  Regularized estimation in sparse high-dimensional time series models , 2013, 1311.4175.

[41]  Murat A. Erdogdu,et al.  Flexible results for quadratic forms with applications to variance components estimation , 2015, 1509.04388.

[42]  M. Rudelson,et al.  Hanson-Wright inequality and sub-gaussian concentration , 2013 .

[43]  P. Robinson.  Gaussian Semiparametric Estimation of Long Range Dependence , 1995 .

[44]  Feng Qi (祁锋).  BOUNDS FOR THE RATIO OF TWO GAMMA FUNCTIONS-FROM WENDEL'S AND RELATED INEQUALITIES TO LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTIONS , 2009, 0904.1048.