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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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References

Akaike, H. (1973) Block Toeplitz matrix inversion. SIAM J. Appl. Math. 24, 234–241.CrossRefGoogle Scholar
Bartlett, M. S. (1966) An Introduction to Stochastic Processes. Cambridge University Press.Google Scholar
Battaglia, F. (1984) Inverse covariances of a multivariate time series. Metron 42, 117–129.Google Scholar
Battaglia, F. (1988) On the estimation of the inverse correlation function. J. Time Series Analysis 9, 1–10.Google Scholar
Battaglia, F. and Bhansali, R. J. (1987) Estimation of the interpolation error variance and an index of linear determinism. Biometrika 74, 771–779.Google Scholar
Baxter, G. (1963) A norm inequality for a ‘finite section’ Wiener-Hopf equation. Illinois J. Math. 7, 97–103.Google Scholar
Berk, K. N. (1974) Consistent autoregressive spectral estimates. Ann. Statist. 2, 489–502.Google Scholar
Besag, J. (1975) Statistical analysis of non-lattice data. The Statistician 24, 179–195.CrossRefGoogle Scholar
Bhansali, R. J. (1980) Autoregressive and window estimates of the inverse correlation function. Biometrika 67, 551–566.CrossRefGoogle Scholar
Bhansali, R. J. (1982) The evaluation of certain quadratic forms occurring in autoregressive model fitting. Ann. Statist. 10, 121–131.CrossRefGoogle Scholar
Bhansali, R. J. (1986) A derivation of the information criteria for selecting autoregressive models. Adv. Appl. Prob. 18, 360–387.CrossRefGoogle Scholar
Brillinger, D. R. (1975) Time Series: Data Analysis and Theory. Holt, New York.Google Scholar
Dempster, A. P. (1972) Covariance selection. Biometrics 28, 157–176.Google Scholar
Friedlander, B., Morf, M., Kailath, T. and Ljung, L. (1979) New inversion formulas for matrices classified in terms of their distance from Toeplitz matrices. Linear Algebra Appl. 27, 31–60.CrossRefGoogle Scholar
Fuller, W. A. and Hasza, D. P. (1981) Properties of predictors for autoregressive time series. J. Amer. Statist. Assoc. 76, 155–161.Google Scholar
Galbraith, R. F. and Galbraith, J. I. (1974) On the inverses of some patterned matrices arising in the theory of stationary time series. J. Appl. Prob. 11, 63–71.Google Scholar
Green, P., Jennison, C. and Seheult, A. (1985) Analysis of field experiments by least-squares smoothing. J. R. Statist. Soc. B 47, 299–315.Google Scholar
Grenander, U. and Szego, G. (1958) Toeplitz Forms and their Applications. University of California Press, Berkeley.Google Scholar
Hammersley, J. M. and Handscomb, D. C. (1965) Monte Carlo Methods. Methuen, London.Google Scholar
Hannan, E. J. (1970) Multiple Time Series. Wiley, New York.Google Scholar
Kailath, T., Vieira, A. and Morf, M. (1978) Inverses of Toeplitz operators, innovations and orthogonal polynomials. SIAM Rev. 20, 106–119.Google Scholar
Kumar, R. (1985) A fast algorithm for solving a Toeplitz system of equations. IEEE Trans. Acoustics, Speech and Signal Processing 33, 254–267.Google Scholar
Kutikov, L. M. (1967) The structure of matrices which are the inverses of the correlation matrices of random vector processes. USSR Comput. Math. and Math. Phys. 7, 58–71.Google Scholar
Lawrance, A. J. and Lewis, P. A. W. (1981) A new autoregressive time series model in exponential variables (NEAR(1)). Adv. Appl. Prob. 13, 826–845.Google Scholar
Masani, P. (1960) The prediction theory of multivariate stochastic processes, III. Acta. Math. 104, 141–162.Google Scholar
Rao, C. R. (1973) Linear Statistical Inference and its Applications. Wiley, New York.Google Scholar
Shaman, P. (1975) An approximate inverse for the covariance matrix of moving average and autoregressive processes. Ann. Statist. 3, 521–538.Google Scholar
Shaman, P. (1976) Approximations for stationary covariance matrices and their inverses with applications to ARMA models. Ann. Statist. 4, 292–301.Google Scholar
Whittle, P. (1951) Hypothesis Testing in Time Series Analysis. Almqvist and Wiksell, Uppsala.Google Scholar
Whittle, P. (1963) Prediction and Regulation by Linear Least-Squares Methods. English University Press, London.Google Scholar
Wise, J. (1955) The autocorrelation function and the spectral density function. Biometrika 42, 151–159.Google Scholar