Covariance matrix estimation for stationary time series

Han Xiao, Wei Biao Wu

Research output: Contribution to journalArticlepeer-review

63 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)466-493
Number of pages28
JournalAnnals of Statistics
Volume40
Issue number1
DOIs
StatePublished - Feb 2012
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

  • Autocovariance matrix
  • Banding
  • Large deviation
  • Physical dependence measure
  • Short range dependence
  • Spectral density
  • Stationary process
  • Tapering
  • Thresholding
  • Toeplitz matrix

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