Open Access
Translator Disclaimer
March, 1976 Approximations for Stationary Covariance Matrices and Their Inverses with Application to ARMA Models
Paul Shaman
Ann. Statist. 4(2): 292-301 (March, 1976). DOI: 10.1214/aos/1176343408

Abstract

Approximation of the covariance matrix $\mathbf{\Sigma}$ of $T$ consecutive observations from a second-order stationary process with continuous positive spectral density $f(\lambda) = \lbrack\sigma^2/(2\pi)^2\rbrack|\sum^\infty_{j=0}\delta_je^{i\lambda j}|^2$ is considered. If $\mathbf{\Sigma}^\ast$ is the covariance matrix corresponding to a process with spectral density $1/\lbrack(2\pi)^2f(\lambda)\rbrack$, then $\mathbf{\Sigma}^\ast - \mathbf{\Sigma}^{-1} \geqq 0$. A matrix $\sigma^{-2}\mathbf{A'A}$ with the property that $\mathbf{\Sigma}^\ast - \sigma^{-2}\mathbf{A'A} - \mathbf{\Sigma}^{-1} \geqq 0$ is also considered. For autoregressive-moving average processes of order $(p, q), \mathbf{\Sigma}^\ast - \sigma^{-2}\mathbf{A'A}$ and $\sigma^{-2}\mathbf{A'A} - \mathbf{\Sigma}^{-1}$ are shown to have rank $\min \lbrack\max (p, q), T\rbrack$ and $\mathbf{\Sigma}^\ast - \mathbf{\Sigma}^{-1}$ to have rank $\min \lbrack 2\max (p, q), T\rbrack$. Some results concerning the covariance determinant are also discussed. If $D_T$ is $\sigma^{-2T}|\mathbf{\Sigma}|$ for sample size $T$ and $D_0 = 1$, then $D_T < D_{T+1}, T = 0, 1, \cdots$, unless the process is autoregressive of order $p$, in which case $1 < D_1 < \cdots < D_p = D_{p+1} = \cdots$.

Citation

Download Citation

Paul Shaman. "Approximations for Stationary Covariance Matrices and Their Inverses with Application to ARMA Models." Ann. Statist. 4 (2) 292 - 301, March, 1976. https://doi.org/10.1214/aos/1176343408

Information

Published: March, 1976
First available in Project Euclid: 12 April 2007

zbMATH: 0326.62062
MathSciNet: MR398033
Digital Object Identifier: 10.1214/aos/1176343408

Subjects:
Primary: 62M10
Secondary: 15A45 , 60G10

Keywords: autoregressive-moving average (ARMA) process , covariance determinant , Covariance matrix , positive semidefinite , rank of matrix , stationary process

Rights: Copyright © 1976 Institute of Mathematical Statistics

JOURNAL ARTICLE
10 PAGES


SHARE
Vol.4 • No. 2 • March, 1976
Back to Top