Qiu, Yumou and Chen, Song Xi (2014): Band Width Selection for High Dimensional Covariance Matrix Estimation. Forthcoming in: Journal of the American Statistical Association

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Abstract
The banding estimator of Bickel and Levina (2008a) and its tapering version of Cai, Zhang and Zhou (2010), are important high dimensional covariance estimators. Both estimators require choosing a band width parameter. We propose a band width selector for the banding covariance estimator by minimizing an empirical estimate of the expected squared Frobenius norms of the estimation error matrix. The ratio consistency of the band width selector to the underlying band width is established. We also provide a lower bound for the coverage probability of the underlying band width being contained in an interval around the band width estimate. Extensions to the band width selection for the tapering estimator and threshold level selection for the thresholding covariance estimator are made. Numerical simulations and a case study on sonar spectrum data are conducted to confirm and demonstrate the proposed band width and threshold estimation approaches.
Item Type:  MPRA Paper 

Original Title:  Band Width Selection for High Dimensional Covariance Matrix Estimation 
English Title:  Band Width Selection for High Dimensional Covariance Matrix Estimation 
Language:  English 
Keywords:  Bandable covariance; Banding estimator; Large $p$, small $n$; Ratioconsistency; Tapering estimator; Thresholding estimator. 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C13  Estimation: General C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C14  Semiparametric and Nonparametric Methods: General C  Mathematical and Quantitative Methods > C5  Econometric Modeling 
Item ID:  59641 
Depositing User:  Professor Song Xi Chen 
Date Deposited:  04 Nov 2014 05:45 
Last Modified:  30 Sep 2019 21:28 
References:  Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis. Wiley, New York. Bai, Z.D., Silverstein, J.W. and Yin, Y.Q. (1998), A Note on the Largest Eigenvalue of a Largedimensional Sample Covariance Matrix. Journal of Multivariate Analysis 26, 166168. Bai, Z.D. and Yin, Y.Q. (1993), Limit of the Smallest Eigenvalue of a Large Dimensional Sample Covariance Matrix. The Annals of Probability 21 12761294. Bickel, P. J. and Levina, E. (2008a), Regularized Estimation of Large Covariance Matrices. The Annals of Statistics 36 199227. Bickel, P. and Levina, E. (2008b), Covariance Regularization by Thresholding. The Annals of Statistics 36 25772604. Cai, T. T., Zhang, C.H. and Zhou, H. (2010), Optimal Rates of Convergence for Covariance Matrix Estimation. The Annals of Statistics 38 21182144. Cai, T. T. and Liu, W. (2011), Adaptive Thresholding for Sparse Covariance Matrix Estimation. Journal of the American Statistical Association 494 672684. Cai, T. T., Liu, W. and Luo, X. (2011), A Constrained $\ell_{1}$ Minimization Approach to Sparse Precision Matrix Estimation. Journal of the American Statistical Association 494 594607. Cai, T. T., Liu, W. and Xia, Y. (2013), TwoSample Covariance Matrix Testing and Support Recovery in HighDimensional and Sparse Settings. Journal of the American Statistical Association 501 265277. Cai, T. and Yuan, M. (2012), Adaptive covariance matrix estimation through block thresholding. The Annals of Statistics 40 20142042. Fan, J., Fan, Y. and Lv, J. (2008), High Dimensional Covariance Matrix Estimation Using a Factor Model. Journal of Econometrics 147 186197. Gorman, R. and Sejnowski, T. (1988a), Analysis of Hidden Units in a Layered Network Trained to Classify Sonar Targets. Neural Networks 1 7589. Gorman, R. and Sejnowski, T. (1988b), Learned Classification of Sonar Targets Using a Massively Parallel Network . IEEE Transactions on Acoustics, Speech and Signal Processing 36 11351140. Hall, P. and Jin, J. (2010), Innovated Higher Criticism for Detecting Sparse Signals in Correlated Noise. The Annals of Statistics 38 16861732. Huang, J., Liu, N., Pourahmadi, M., and Liu, L.} (2006), Covariance Matrix Selection and Estimation via Penalised Normal Likelihood. Biometrika 93 8598. Jing, B.Y., Shao, Q.M. and Wang, Q.Y. (2003), Selfnormalized Cramer Type Large Deviations for Independent Random Variables. The Annals of Probability 31 21672215. Johnstone, I. (2001), On the Distribution of the Largest Eigenvalue in Principal Components Analysis. \textit{The Annals of Statistics 29 295327. Levina, E., Rothman, A. and Zhu, J. (2008), Sparse Estimation of Large Covariance Matrices Via a Nested Lasso Penalty. The Annals of Applied Statistics 2 245263. Qiu, Y. and Chen, S. (2012), Test for Bandedness of Highdimensional Covariance Matrices and Bandwidth Estimation. The Annals of Statistics 40 12851314. Rothman, A. J., Levina, E. and Zhu, J. (2009), Generalized Thresholding of Large Covariance Matrices. Journal of the American Statistical Association 104 177186. Rothman, A. J., Levina, E. and Zhu, J. (2010), A new approach to Choleskybased covariance regularization in high dimensions. Biometrika 97 539550. van der Vaart, A.W.} (2000), Asymptotic Statistics. Cambridge University Press, Cambridge. Wu, W. B. and Pourahmadi, M. (2003), Nonparametric Estimation of Large Covariance Matrices of Longitudinal Data. Biometrika 93, 831844. Xue, L. and Zou, H. (2012), Regularized Rankbased Estimation of Highdimensional Nonparanormal Graphical Models. The Annals of Statistics 40 25412571. Xue, L. and Zou, H. (2013), Minimax Optimal Estimation of General Bandable Covariance Matrices. Journal of Multivariate Analysis 116 4551. Yi, F. and Zou, H. (2013), SUREtuned Tapering Estimation of Large Covariance Matrices. Computational Statistics and Data Analysis 58 339351. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/59641 