Abstract
The problem of estimation of nonlinear time series models which are a composition of nonlinear elements and linear stochastic processes is considered. The compositions studied include the cascade and parallel connections. The problem of nonparametric estimation of underlying nonlinearities is examined. It is resolved by solving Fredholm’s integral equations of the second kind arising in the estimation problem. As a result, the nonparametric orthogonal series estimates of nonlinearities are derived and their asymptotic as well as some small sample properties are established.
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References
Andrews, D.W.K. (1984), ‘Non-strong mixing autoregressive processes’, J. Appl. Prob., 21, 930–934.
Andrews, D.W.K. (1985), ‘A nearly independent, but non-strong mixing, triangular array’, J. Appl. Prob., 22, 729–731.
Bierens, H.T. (1983), ‘Uniform consistency of kernel estimators of a regression function under generalized conditions’, J. Amer. Statist. Assoc., 78, 699–707.
Breiman, L. and Friedman, J.H. (1985), ‘Estimating optimal transformations for multiple regression and correlation’, J. Amer. Statist. Assoc., 80, 580–597.
Brillinger, D.R. (1977), ‘The identification of a particular nonlinear timer series system’, Biometrika, 64, 509–515.
Brillinger, D.R. (1980), Time Series. Data Analysis and Theory, Holden-Day, Inc., San Francisco, Expanded Edition.
Collomb, G. and Härdle, W. (1985), ‘Strong uniform convergence rates in robust nonparametric time series analysis and prediction: kernel regression estimation from dependent observations’, Stochastic Process.Appl., 23, 77–81.
Eubank, R.L. (1980), Spline Smoothing and Nonparametric Regression, Marcel Dekker, New York.
Georgiev, A. (1984), ‘Nonparametric system identification by kernel methods’, IEEE Trans. Automat. Control, AC-29, 356–358.
Greblicki, W. and Pawlak, M. (1986), ‘Identification of discrete Hammerstein systems using kernel regression estimates’, IEEE Trans. Automat. Control, AC-31, 74–77.
Greblicki, W. and Pawlak, M. (1989), ‘Recursive nonparametric identification of Hammerstein systems’, J. of Franklin Institute, 326, 461–481.
Greblicki, W. and Pawlak, M. (1989), ‘Nonparametric identification of Hammerstein systems’, IEEE Trans. Inform. Theory, 35, 409–418.
Greblicki, W. and Pawlak, M. (1984), ‘Hermite series estimates of a probability density and its derivatives’, J. Multivariate Anal. 15, 176–182.
Greblicki, W. and Pawlak, M. (1985), ‘Fourier and Hermite series estimates of regression functions’, Ann. Inst. Statist. Math., 37, 443–459.
Györfi, L., Härdle, W., Sarda, P. and Vieu, P. (1989), Nonparametric Curve Estimation from Time Series. Springer-Verlag, Berlin.
Hall, P. (1987), ‘Cross-validation and the smoothing of orthogonal series density estimators’, J. Multivariate Anal., 21, 189–206.
Hunter, I.W. and Korenberg, M.J. (1986), ‘The identification of nonlinear biological systems: Wiener and Hammerstein cascade models’, Biol. Cybern., 55, 135–144.
Krzyzak, A. (1990), ‘On estimation of a class of nonlinear systems by the kernel regression estimate’, IEEE Trans. Inform. Theory, IT-36, 141–152.
Lancaster, H.O. (1958), ‘The structure of bivariate distributions’, Ann. Math. Statist., 29, 719–736.
O’Leary, D.P. (1985), ‘Identification of sensory systems and neural systems’, IFAC Symposium on Identification and System Parameter Estimation, Pergamon Press, 77–84.
Pawlak, M. (1988), ‘Identification of a class of parallel systems’, The 26th Allerton Conf. in Communication, Control, and Computing, I, 366–367.
Pawlak, M. (1990), ‘Estimation of a class of additive nonlinear time series models’, Manuscript in preparation.
Robinson, P.M. (1986), ‘On the consistency and finite-sample properties on nonparametric kernel time series regression, autoregression and density estimators’, Ann. Inst. Statist. Math., Part A, 38, 539–549.
Stone, C.T. (1985), ‘Additive regression and other nonparametric models’, Ann. Statist., 13, 689–705.
Stone, C.J. (1977), ‘Consistent nonparametric regression’, Ann. Statist., 5, 595–645.
Wise, G.L. and Thomas, J.B. (1975), ‘A characterization of Markov sequences’, J. Franklin Institute, 229, 269–278.
Yakowitz, S. (1985), ‘Nonparametric density estimation prediction and regression for Markov sequences’, J. Amer. Statist. Assoc., 80, 215–221.
Tricomi, F.G. (1985), Integral Equations, Dover, New York.
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© 1991 Springer Science+Business Media Dordrecht
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Pawlak, M., Greblicki, W. (1991). Nonparametric Estimation of a Class of Nonlinear Time Series Models. In: Roussas, G. (eds) Nonparametric Functional Estimation and Related Topics. NATO ASI Series, vol 335. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3222-0_40
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DOI: https://doi.org/10.1007/978-94-011-3222-0_40
Publisher Name: Springer, Dordrecht
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