Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T18:25:54.375Z Has data issue: false hasContentIssue false
  • English
  • Français

Separable lower triangular bilinear model

Part of: Stochastic processes Inference from stochastic processes

Published online by Cambridge University Press:  14 July 2016

Hai-Bin Wang  and
Bo-Cheng Wei
Hai-Bin Wang*
Affiliation:
Southeast University, Nanjing, and Xiamen University
Bo-Cheng Wei*
Affiliation:
Southeast University, Nanjing
*
∗ Postal address: Department of Mathematics, Xiamen University, Xiamen 361005, P. R. China. Email address: [email protected]
∗∗ Postal address: Southeast University, Nanjing, Si Pai Lou 2#, Nanjing 210096, P. R. China.
Get access Rights & Permissions [Opens in a new window]

Abstract

The aim of this paper is to analyze the probabilistic structure for a rather general class of bilinear models systematically. First, the sufficient and necessary conditions for stationarity are given with a concise expression. Then both the autocovariance function and the spectral density function are obtained. The Yule–Walker-type difference equations for autocovariances are derived by means of the spectral density function. Concerning the second-order probabilistic structure, the model is similar to an ARMA model. The third-order probabilistic structure for the model is discussed and a group of Yule–Walker-type difference equations for third-order cumulants are discovered.

Keywords

Time seriesbilinear modelstationarityautocovariancespectral densitythird-order cumulantsbispectral densityYule–Walker equations

MSC classification

Primary: 62M10: Time series, auto-correlation, regression, etc.
Secondary: 60G10: Stationary processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 1 , March 2004 , pp. 221 - 235
Copyright
Copyright © Applied Probability Trust 2004 

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

Granger, C. W. J., and Anderson, A. P. (1978). An Introduction to Bilinear Time Series Models. Vandenhoeck and Ruprecht, Gottingen.Google Scholar
Liu, J., and Brockwell, P. J. (1988). On the general bilinear time series model. J. Appl. Prob. 25, 553–564.CrossRefGoogle Scholar
Liu, S.-I. (1985). Theory of bilinear time series models. Commun. Statist. Theory Meth. 14, 2549–2561.CrossRefGoogle Scholar
Rao, M. B., Subba Rao, T., and Walker, A. M. (1983). On the existence of some bilinear time series models. J. Time Ser. Anal. 4, 95–110.Google Scholar
Sesay, S. A. O., and Subba Rao, T. (1988). Yule–Walker type difference equations for higher-order moments and cumulants for the bilinear time series models. J. Time Ser. Anal. 9, 385–401.Google Scholar
Sesay, S. A. O., and Subba Rao, T. (1991). Difference equations for higher-order moments and cumulants for the bilinear time series model. J. Time Ser. Anal. 12, 159–177.CrossRefGoogle 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 nonlinear stationary time series with reference to bilinear time series models. In Handbook of Statistics, Vol. 3, Time Series in the Frequency Domain, eds Brillinger, D. R. and Krishnaiah, P. R., North-Holland, Amsterdam, pp. 293–319.Google Scholar
Subba Rao, T., and Gabr, M. (1984). An Introduction to Bispectral Analysis and Bilinear Time Series Models (Lecture Notes Statist. 24). Springer, New York.Google Scholar
Terdik, G. (1985). Transfer functions and conditions for stationarity of bilinear models with Gaussian residuals. Proc. R. Soc. London A 400, 315–330.Google Scholar
Terdik, G., and Subba Rao, T. (1989). On Wiener–Ito representation and the best linear prediction for bilinear time series. J. Appl. Prob. 26, 274–286.Google Scholar