Qiu, Yumou and Chen, Songxi (2012): Test for Bandedness of High Dimensional Covariance Matrices with Bandwidth Estimation. Published in:

PDF
MPRA_paper_46242.pdf Download (484kB)  Preview 
Abstract
Motivated by the latest effort to employ banded matrices to estimate a highdimensional covariance Σ , we propose a test for Σ being banded with possible diverging bandwidth. The test is adaptive to the “large p , small n ” situations without assuming a specific parametric distribution for the data. We also formulate a consistent estimator for the bandwidth of a banded highdimensional covariance matrix. The properties of the test and the bandwidth estimator are investigated by theoretical evaluations and simulation studies, as well as an empirical analysis on a protein mass spectroscopy data.
Item Type:  MPRA Paper 

Original Title:  Test for Bandedness of High Dimensional Covariance Matrices with Bandwidth Estimation 
Language:  English 
Keywords:  Banded covariance matrix,Bandwidth estimation,High data dimension,Large p,small n,Nonparametric. 
Subjects:  C  Mathematical and Quantitative Methods > C0  General C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General C  Mathematical and Quantitative Methods > C2  Single Equation Models ; Single Variables C  Mathematical and Quantitative Methods > C3  Multiple or Simultaneous Equation Models ; Multiple Variables C  Mathematical and Quantitative Methods > C4  Econometric and Statistical Methods: Special Topics C  Mathematical and Quantitative Methods > C5  Econometric Modeling C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling C  Mathematical and Quantitative Methods > C7  Game Theory and Bargaining Theory C  Mathematical and Quantitative Methods > C8  Data Collection and Data Estimation Methodology ; Computer Programs C  Mathematical and Quantitative Methods > C9  Design of Experiments G  Financial Economics > G0  General 
Item ID:  46242 
Depositing User:  Professor Songxi Chen 
Date Deposited:  17 Apr 2013 10:04 
Last Modified:  04 Oct 2019 02:54 
References:  [1] Adam, B.L., Qu, Y, Davis, J. W, Ward, M. D., Clements, M. A., Cazares, L.H., Semmes, O. J., Schellhamm, P F., Yasui, Y, Feng, Z. and Wright, Jr, G. L.W(2003). Serum Protein Fingerprinting Coupled With a Patternmatching Algorithm Distinguishes Prostate Cancer from Benign Prostate Hyperplasia and Healthy mean.Cancer Research 63 36093614. [2] Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis. Wiley,New York. [3] Bai, Z. and Saranadasa, H. (1996). E�ect of High Dimension: by an Example of a Two Sample Problem.Statistica Sinica 6 311329. [4] Bai, Z. D. and Silverstein, J.W. (2005). Spectral Analysis of Large Dimensional Random Matrices. Scienti�c Press, Beijing. [5] Bai, Z.D., Silverstein, J.W. and Yin, Y.Q. (1988). A Note on the Largest Eigenvalue of a Largedimensional Sample Covariance Matrix. Journal of Multivariate Analysis 26 166168. [6] 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. [7] Bickel, P. J. and Levina, E. (2008a). Regularized Estimation of Large Covariance Matrices. The Annals of Statistics 36 199227. [8] Bickel, P. and Levina, E. (2008b). Covariance Regularization by Thresholding. The Annals of Statistics 36 25772604. [9] Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley, New York. [10] Cai, T. and Jiang T. (2011). Limiting Laws of Coherence of Random Matrices with Applications to Testing Covariance Structure and Construction of Compressed Sensing Matrices. The Annals of Statistics 39 14961525. [11] Cai, T. and Jiang T. (2011). Phase Transition in Limiting Distributions of Coherence of HighDimensional Random Matrices. technical report. [12] Cai, T., Zhang, C.H. and Zhou, H. (2010). Optimal Rates of Convergence for Covariance Matrix Estimation. The Annals of Statistics 38 21182144. [13] Chen, S. X., Zhang L.X. and Zhong, P.S. (2010).Tests for High Dimensional Covariance Matrices. Journal of the American Statistical Association 105 810819. [14] Cleveland, W. and Devlin, S. J. (1988). Locally Weighted Regression: An Approach to Regression Analysis by Local Fitting. Journal of American Statistical Association 83 596610. [15] El Karoui, N. (2011). On the largest eigenvalue of Wishart matrices with identity covariance when n, p and n/p tend to in�nity. manuscript. [16] Fan, J., Fan, Y. and Lv, J. (2008). High Dimensional Covariance Matrix Estimation Using a Factor Model. Journal of Econometrics 147 186197. [17] Fan, J. and Gijbles, I. (1996). Local Polynomial Smoothing. Chapman and Hall,London. [18] Huang, J., Liu, N., Pourahmadi, M., and Liu, L.(2006). Covariance Matrix Selection and Estimation via Penalised Normal Likelihood. Biometrika 93 8598. [19] Jiang, T. (2004). The Asymptotic Distributions of the Largest Entries of Sample Correlation Matrices. The Annals of Applied Probability 14 865880. [20] John, S. (1972). The Distribution of a Statistic Used for Testing Sphericity of Normal Distributions. Biometrika 59 169173. [21] Johnstone, I. M. (2001). On the Distribution of the Largest Eigenvalue in Principal Components Analysis. The Annals of Statistics 29 295327. [22] Ledoit, O. and Wolf, M. (2002). Some Hypothesis Tests for the Covariance Matrix When the Dimension is Large Compare to the Sample Size. The Annals of Statistics 30 10811102. [23] Levina, E., Rothman, A. and Zhu, J. (2008). Sparse Estimation of Large Covariance Matrices Via a Nested Lasso Penalty. The Annals of Applied Statistics 2245263. [24] Liu, W.D., Lin, Z. and Shao, Q.M. (2008). The Asymptotic Distribution and BerryEssen Bound of a New Test for Independence in High Dimension with an Application to Stochastic Optimization. The Annals of Applied Probability 18 23372366. [25] Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. Wiley, New York. [26] Nagao, H. (1973). On Some Test Criteria for Covariance Matrix. The Annals of Statistics 1 700709. [27] Rothman, A. J., Levina, E. and Zhu, J. (2009). Generalized Thresholding of Large Covariance Matrices. Journal of the American Statistical Association 104 177186. [28] Rothman, A. J., Levina, E. and Zhu, J. (2010). A new approach to Choleskybased covariance regularization in high dimensions. Biometrika 97 539550. [29] Schott, J. R. (2005). Testing for Complete Independence in High Dimensions.Biometrika 92 951956. [30] Tibshirani, R., Saunders, M., Rosset, S., Zhu, J. and Knight, K. (2005).Sparsity and Smoothness via the Fused Lasso. Journal of Royal Statistical Society Ser. B 67 91108. [31] Wagaman, A. S. and Levina, E. (2009). Discovering Sparse Covariance Structures With the Isomap. Journal of Computational and Graphical Statistics 18 551572. [32] Wu, W. B. and Pourahmadi, M. (2003). Nonparametric Estimation of Large Covariance Matrices of Longitudinal Data. Biometrika 93 831844. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/46242 