Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-28T13:19:28.858Z Has data issue: false hasContentIssue false
  • English
  • Français

On Wiener–Ito representation and the best linear predictors for bilinear time series

Published online by Cambridge University Press:  14 July 2016

Gy. Terdik  and
T. Subba Rao
Gy. Terdik*
Affiliation:
Kossuth Lajos University, Debrecen
T. Subba Rao*
Affiliation:
UMIST
*
∗ Postal address: Mathematical Institute, KLTM, Kossuth L. University, Pf 12 Debrecen, Hungary.
∗∗ Postal address: Department of Mathematics, UMIST, PO Box 88, Manchester M60 1QD, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper gives the necessary and sufficient condition for the second-order stationarity of lower triangular bilinear models; this condition is in terms of norms of higher-order transfer functions. The spectral density function of the process is evaluated in terms of the transfer functions, and it is shown that the second-order structure is similar to a linear ARMA model with uncorrelated errors. The best linear predictor is obtained and it is shown that the variance of the prediction error is always greater than the optimal prediction error variance obtained from the bilinear model. An expression for the bispectral density function is also obtained.

Keywords

BILINEAR MODELSHIGHER-ORDER TRANSFER FUNCTIONSSPECTRALINEAR PREDICTORSSTATIONARITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 2 , June 1989 , pp. 274 - 286
Copyright
Copyright © Applied Probability Trust 1989 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Brillinger, D. R. (1965) An introduction to polyspectra. Ann. Math. Statist. 36, 1351–1374.Google Scholar
Dobrushin, R. L. (1979) Gaussian and their subordinated generalised fields. Ann. Prob. 7, 1–28.CrossRefGoogle Scholar
Granger, C. W. J. and Andersen, A. (1978) On the invertibility of time series models. Stoch. Proc. Appl. 8, 87–92.CrossRefGoogle Scholar
Hannan, E. J. (1982) A note on bilinear time series models. Stoch. Proc. Appl. 12, 221–224.Google Scholar
Major, P. (1981) Multiple Wiener–Ito Integrals. Lecture Notes in Mathematics 849, Springer-Verlag, New York.CrossRefGoogle Scholar
Priestley, M. B. (1981) Spectral Analysis and Time Series. Academic Press, London.Google Scholar
Subba Rao, T. (1977) On the estimation of bilinear time series models. Bull. Internat. Statist. Inst. 41, 413–417.Google Scholar
Subba Rao, T. (1981) On the theory of bilinear time series models. J. R. Statist. Soc. B 43, 244–255.Google Scholar
Subba Rao, T. (1983) The bispectral analysis of non-linear stationary time series with reference to bilinear time series models. Handbook of Statistics, Vol. 3, North Holland, 293–319.Google Scholar
Terdik, G. (1985) Transfer functions and conditions for stationarity of bilinear models with Gaussian residuals. Proc. Roy. Soc. London A 400, 315–330.Google Scholar