The Annals of Statistics

Covariance matrix estimation for stationary time series

Han Xiao and Wei Biao Wu

Full-text: Open access

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.

Article information

Source
Ann. Statist., Volume 40, Number 1 (2012), 466-493.

Dates
First available in Project Euclid: 16 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.aos/1334581750

Digital Object Identifier
doi:10.1214/11-AOS967

Mathematical Reviews number (MathSciNet)
MR3014314

Zentralblatt MATH identifier
1246.62191

Subjects
Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62H12: Estimation

Keywords
Autocovariance matrix banding large deviation physical dependence measure short range dependence spectral density stationary process tapering thresholding Toeplitz matrix

Citation

Xiao, Han; Wu, Wei Biao. Covariance matrix estimation for stationary time series. Ann. Statist. 40 (2012), no. 1, 466--493. doi:10.1214/11-AOS967. https://projecteuclid.org/euclid.aos/1334581750


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