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On a relationship between the inverse of a stationary covariance matrix and the linear interpolator

Published online by Cambridge University Press:  14 July 2016

R. J. Bhansali
R. J. Bhansali*
Affiliation:
University of Liverpool
*
∗Postal address: Department of Statistics and Computational Mathematics, University of Liverpool, Victoria Building, Brownlow Hill, P.O. Box 147, Liverpool L69 3BX, UK.
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Abstract

Let {xt} be a discrete-time multivariate stationary process possessing an infinite autoregressive representation and let ΓB(k), ΓF(k) and Γ be the block Toeplitz covariance matrices of xB(k) = [x′–1, x′–2, · ··, x′–k]′, xF(k) = [x′1, x′2 · ·· x′k] and x = [·· ·x′–2, x′–1, x′0, x′1, x′2 · ··]′ respectively, where k ≧ 1, is finite or infinite. Also let φ m,n(j) and δm,n(u) be the coefficients of xt+ j and xt– u respectively in the linear least-squares interpolator of xt from xt+ 1, · ··, xt+ m; xt− 1, · ··, xt– n, where m, n ≧ 0, 0 ≦ j ≦ m, 0 ≦ u ≦ n are integers, zt(m, n) denote the interpolation error and τ2(m, n) = E[zt(m, n)zt(m, n)′]. A physical interpretation for the components of ΓB(k)–1, ΓF(k)–1 and Γ–1 is given by relating these components to the φm,n(j) δm,n(u) and τ2(m, n). A similar result is shown to hold also for the estimators of ΓB(k)–l and the interpolation parameters when these have been obtained from a realization of length T of {xt}. Some of the applications of the results are considered.

Keywords

BLOCK TOEPLITZ MATRIXINVERSE CORRELATION FUNCTIONCHOLESKY DECOMPOSITIONBACKWARD AND FORWARD AUTOREGRESSIVE REPRESENTATIONSLINEAR PROCESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 27 , Issue 1 , March 1990 , pp. 156 - 170
Copyright
Copyright © Applied Probability Trust 1990 

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