Covariance matrix estimation for stationary time series
Han Xiao and Wei Biao Wu
Source: Ann. Statist. Volume 40, Number 1
(2012), 466-493.
Abstract
We obtain a sharp convergence rate for banded covariance matrix estimates of stationary processes. A precise order of magnitude is derived for spectral radius of sample covariance matrices. We also consider a thresholded covariance matrix estimator that can better characterize sparsity if the true covariance matrix is sparse. As our main tool, we implement Toeplitz [Math. Ann. 70 (1911) 351–376] idea and relate eigenvalues of covariance matrices to the spectral densities or Fourier transforms of the covariances. We develop a large deviation result for quadratic forms of stationary processes using m-dependence approximation, under the framework of causal representation and physical dependence measures.
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Keywords: Autocovariance matrix; banding; large deviation; physical dependence measure; short range dependence; spectral density; stationary process; tapering; thresholding; Toeplitz matrix
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