BANDING SAMPLE AUTOCOVARIANCE MATRICES OF STATIONARY PROCESSES

We consider estimation of covariance matrices of stationary processes. Under a short-range dependence condition for a wide class of nonlinear processes, it is shown that the banded covariance matrix estimates converge in operator norm to the true covariance matrix with explicit rates of convergence. We also establish the consistency of the estimate of the inverse covariance matrix. These results are applied to a prediction problem, and error bounds for the finite predictor coeffi- cients are obtained. A sub-sampling approach is proposed to choose the banding parameter, and simulation results reveal its satisfactory performance for linear and certain nonlinear processes as the procedure is solely based on the second-order characteristics of the underlying process. Selection of the band parameter for non- linear processes remains an open problem.

[1]  J. Doob.  Stochastic processes , 1953 .

[2]  U. Grenander,et al.  Toeplitz Forms And Their Applications , 1958 .

[3]  K. Berk.  Consistent Autoregressive Spectral Estimates , 1974 .

[4]  E. J. Hannan,et al.  The central limit theorem for time series regression , 1979 .

[5]  Richard A. Davis,et al.  Simple consistent estimation of the coefficients of a linear filter , 1988 .

[6]  E. Hannan,et al.  The statistical theory of linear systems , 1989 .

[7]  H. Tong.  Non-linear time series. A dynamical system approach , 1990 .

[8]  Richard A. Davis,et al.  Time Series: Theory and Methods , 2013 .

[9]  Richard A. Davis,et al.  Time Series: Theory and Methods (2nd ed.). , 1992 .

[10]  I. Johnstone.  On the distribution of the largest eigenvalue in principal components analysis , 2001 .

[11]  Arnold J Stromberg,et al.  Subsampling , 2001, Technometrics.

[12]  E. Parzen.  Foundations of Time Series Analysis and Prediction Theory , 2002 .

[13]  Ching-Kang Ing,et al.  On same-realization prediction in an infinite-order autoregressive process , 2003 .

[14]  A. Dembo,et al.  Spectral measure of large random Hankel, Markov and Toeplitz matrices , 2003, math/0307330.

[15]  M. Pourahmadi,et al.  Nonparametric estimation of large covariance matrices of longitudinal data , 2003 .

[16]  W. Wu,et al.  Nonlinear system theory: another look at dependence. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[17]  J. S. Marron,et al.  Geometric representation of high dimension, low sample size data , 2005 .

[18]  R. Carroll,et al.  Nonparametric estimation of correlation functions in longitudinal and spatial data, with application to colon carcinogenesis experiments , 2007, 0710.3638.

[19]  Wei Biao Wu,et al.  STRONG INVARIANCE PRINCIPLES FOR DEPENDENT RANDOM VARIABLES , 2007, 0711.3674.

[20]  Noureddine El Karoui.  Tracy–Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices , 2005, math/0503109.

[21]  X. Shao,et al.  Asymptotic spectral theory for nonlinear time series , 2006, math/0611029.

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

[23]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.