The Annals of Statistics
- Ann. Statist.
- Volume 40, Number 1 (2012), 466-493.
Covariance matrix estimation for stationary time series
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.
Ann. Statist., Volume 40, Number 1 (2012), 466-493.
First available in Project Euclid: 16 April 2012
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62H12: Estimation
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
- Supplementary material: Additional technical proofs. We give the proofs of Remark 5 and Lemma 9, as well as a few remarks on Lemma 9.