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The exact bispectra for bilinear realizable processes with hermite degree 2

Published online by Cambridge University Press:  01 July 2016

György Terdik*
Affiliation:
University of Arkansas
Laurie Meaux*
Affiliation:
University of Arkansas
*
Present address: Institutum Mathematicum Universitatis Debreceniensis, H-4010 Debrecen Pf. 12, Hungary.
∗∗Postal address: Department of Mathematical Sciences, University of Arkansas, SE 301, Fayetteville, AR 72701, USA.

Abstract

This paper deals with the stationary bilinear model with Hermite degree 2 in discrete time which is built up by the first- and second-order Hermite polynomial of a Gaussian white noise process. The exact spectrum and bispectrum is constructed in terms of the transfer functions of the model.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1991 

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References

Brillinger, D. R. (1965) An introduction to polyspectra. Ann. Math. Statist. 36, 13511374.Google Scholar
Brillinger, D. R. (1970) The identification of polynomial systems by means of higher order spectra. J Sound Vib. 12, 301314.CrossRefGoogle Scholar
Brillinger, D. R. (1985) Fourier inference: Some methods for the analysis of array and nongaussian series data. Water Resources Bull. 21, 743756.Google Scholar
Brillinger, D. R. (1989) Parameter estimation for nongaussian processes via second and third order spectra with an application to some endocrine data. Adv. Meth. Physiol. Syst. Mod. Plenum, New York.Google Scholar
Brillinger, D. R. and Rosenblatt, M. (1967) Asymptotic theory of estimates of kth order spectra. In Spectral Analysis of Time Series , ed. Harris, B., pp. 153188. Wiley, New York.Google Scholar
Brockett, P. L., Hinich, M. J. and Patterson, D. (1988) Bispectral-based tests for the detection of gaussianity and linearity in time series. J. Amer. Statist. Assoc. , 83, 657664.Google Scholar
Major, P. (1981) Multiple Wiener–Ito Integrals. In Lecture Notes in Mathematics 849, Springer-Verlag, New York.Google Scholar
Newton, H. J. (1988) Timeslab: A Time Series Analysis Laboratory. Wadsworth & Brooks/Cole, Pacific Grove, CA.Google Scholar
Priestley, M. B. (1978) Nonlinear models in time series analysis. The Statistician 27, 159–76.Google Scholar
Rudin, W. (1973) Functional Analysis. McGraw-Hill, New York.Google Scholar
Rugh, W. J. (1981) Nonlinear System Theory. Johns Hopkins University Press, Baltimore, MD.Google Scholar
Subba Rao, T. and Gabr, M. (1984) An Introduction to Bispectral Analysis and Bilinear Time Series Models. Lecture Notes in Statistics, Springer-Verlag, New York.Google Scholar
Terdik, Gy. (1985) Transfer functions and conditions for stationarity of bilinear models with Gaussian residuals. Proc. R. Soc. London A 400, 315330.Google Scholar
Terdik, Gy. (1988) Generalized Hermite polynomials and estimation of kernels for discrete I/O Wiener models. Prob. Control Inf. Theory 17, 4961.Google Scholar
Terdik, Gy. (1989) Bilinear state space realization for polynomial systems. Technical Report , Dept, of Statistics, University of California, Berkeley, CA 94720.Google Scholar
Terdik, Gy. and Subba Rao, T. (1989) On Wiener–Ito representation and the best linear predictors for bilinear time series. J. Appl. Prob. 26, 274286.Google Scholar