Abstract
The present paper treats the identification of parametric nonminimum phase transfer function. We propose a method of identification based on the inner outer factorization of stable transfer function. It consists of identifying the outer and inner parts of a transfer function separately. The outer part is identified by the use of the second-order spectral estimate from the observed linear process, while the inner part is identified by the use of a higher-order cumulant spectral estimate from the observed process. Respective parameter estimators are determined in the light of asymptotic efficiency. In order to estimate the order of the inner part of a transfer function, a criterion is proposed. It is introduced based on the same principle as in the case of Akaike's AIC.
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References
Akaike, H. (1974). A new look at the statistical model identification, IEEE Trans. Automat. Control, 19, 716–723.
Åström, K. J. and Söderström, T. (1974). Uniqueness of the maximum likelihood estimates of the parameters of an ARMA model, IEEE Trans. Automat. Control, 19, 769–773.
Box, G. E. P. and Jenkins, G. M. (1976). Time Series Analysis, Forecasting and Control, Holden-Day, San Francisco.
Brillinger, D. R. (1975). Time Series: Data Analysis and Theory, Holt, New York.
Brillinger, D. R. and Rosenblatt, M. (1967). Asymptotic theory of estimates of k-th order spectra, Spectral Analysis of Time Series (ed. B. Harris), 153–188, Wiley, New York.
Duren, P. L. (1970). Theory of H p Spaces, Academic Press, New York.
Giannakis, G. B. and Swami, A. (1990). On estimating noncausal nonminimum phase ARMA models of non-Gaussian processes, IEEE Trans. Acoust. Speech Signal Process., 38, 478–495.
Hannan, E. J. (1973). The asymptotic theory of linear time-series models, J. Appl. Probab., 10, 130–145.
Hannan, E. J. (1980). The estimation of the order of an ARMA process, Ann. Statist., 8, 1071–1081.
Hosoya, Y. and Taniguchi, M. (1982). A central limit theorem for stationary processes and the parameter estimation of linear processes, Ann. Statist., 10, 132–153.
Ibragimov, I. A. (1967). On maximum likelihood estimation of parameters of the spectral density of stationary time series, Theory Probab. Appl., 120, 115–119.
Kabaila, P. (1980). An optimality property of the least-squares estimate of the parameter of the spectrum of a purely nondeterministic time series, Ann. Statist., 8, 1082–1092.
Keenan, D. M. (1985). Asymptotic properties of minimization estimators for time series parameters, Ann. Statist., 13, 369–382.
Keenan, D. M. (1987). Limiting behavior of functionals of higher-order sample cumulant spectra, Ann. Statist., 15, 134–151.
Lii, K. S. and Rosenblatt, M. (1982). Deconvolution and estimation of transfer function phase and coefficients for non-Gaussian linear processes, Ann. Statist., 10, 1195–1208.
Rosenblatt, M. (1980). Linear processes and bispectra, J. Appl. Probab., 17, 265–270.
Taniguchi, M. (1981). An estimation procedure of parameters of a certain spectral density model, J. Roy. Statist. Soc. Ser. B, 43, 34–40.
Tugnait, J. K. (1986). Identification of nonminimum phase linear stochastic systems, Automatica J. IFAC, 22, 457–464.
Walker, A. M. (1964). Asymptotic properties of least squares estimates of parameters of the spectrum of a stationary nondeterministic time series, J. Austral. Math. Soc., 4, 363–384.
Whittle, P. (1953). Estimation and information in stationary time series, Ark. Math. Astr. Fys., 2, 423–434.
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Kumon, M. Identification of non-minimum phase transfer function using higher-order spectrum. Ann Inst Stat Math 44, 239–260 (1992). https://doi.org/10.1007/BF00058639
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DOI: https://doi.org/10.1007/BF00058639