Covariance and precision matrix estimation for high-dimensional time series

Abstract

We consider estimation of covariance matrices and their inverses (a.k.a. precision matrices) for high-dimensional stationary and locally stationary time series. In the latter case the covariance matrices evolve smoothly in time, thus forming a covariance matrix function. Using the functional dependence measure of Wu [Proc. Natl. Acad. Sci. USA 102 (2005) 14150-14154 (electronic)], we obtain the rate of convergence for the thresholded estimate and illustrate how the dependence affects the rate of convergence. Asymptotic properties are also obtained for the precision matrix estimate which is based on the graphical Lasso principle. Our theory substantially generalizes earlier ones by allowing dependence, by allowing nonstationarity and by relaxing the associated moment conditions.

Published in at http://dx.doi.org/10.1214/13-AOS1182 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Regularized estimation of large covariance matrices
• Covariance regularization by thresholding
• Sparse permutation invariant covariance estimation
• Optimal rates of convergence for covariance matrix estimation
• Network exploration via the adaptive LASSO and SCAD penalties
• Optimal rates of convergence for sparse covariance matrix estimation

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