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Some ARMA models for dependent sequences of poisson counts

  • Ed Mckenzie (a1)


A family of models for discrete-time processes with Poisson marginal distributions is developed and investigated. They have the same correlation structure as the linear ARMA processes. The joint distribution of n consecutive observations in such a process is derived and its properties discussed. In particular, time-reversibility and asymptotic behaviour are considered in detail. A vector autoregressive process is constructed and the behaviour of its components, which are Poisson ARMA processes, is considered. In particular, the two-dimensional case is discussed in detail.


Corresponding author

Postal address: Department of Mathematics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, UK.


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Al-Osh, M. A. and Alzaid, A. A. (1987) First order integer-valued autoregressive (INAR(1)) process. J. Time Series Anal. 8, 261275.
Box, G. E. P. and Jenkins, G. M. (1970) Time series Analysis: Forecasting and Control. Holden-Day, San Francisco.
Brillinger, D. and Rosenblatt, M. (1967) Asymptotic theory of estimates of kth order spectra. In Spectral Analysis of Time Series, (ed. by Harris, B.), Wiley, New-York, 153188.
Chiang, C. L. (1980) An Introduction to Stochastic Processes and their Applications. Krieger, Huntington, NY.
Dwass, M. and Teicher, H. (1957) On infinitely divisible random vectors. Ann. Math. Statist. 28, 461470.
Gaver, D. P. and Lewis, P. A. W. (1980) First order autoregressive gamma sequences and point processes. Adv. Appl. Prob. 12, 727745.
Griffiths, R. C., Milne, R. K. and Wood, R. (1979) Aspects of correlation in bivariate Poisson distributions and processes. Austral. J. Statist. 21, 238255.
Jacobs, P. A. and Lewis, P. A. W. (1978a) Discrete time series generated by mixtures I; correlational and runs properties. J. R. Statist. Soc. B 40, 94105.
Jacobs, P. A. and Lewis, P. A. W. (1978b) Discrete time series generated by mixtures II: Asymptotic properties. J. R. Statist. Soc. B 40, 222228.
Jacobs, P. A. and Lewis, P. A. W. (1983) Stationary discrete autoregressive moving average time series generated by mixtures J. Time Series Anal. 4, 1936.
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.
Lawrance, A. J. and Lewis, P. A. W. (1980) The exponential autoregressive moving average EARMA(p, q) process. J. R. Statist. Soc. B42, 150161.
Mckendrick, A. G. (1962) The application of mathematics to medical problems. Proc. Edinburgh. Math. Soc. XLIV, 98130.
Mckenzie, E. (1985) Some simple models for discrete variate time series. Water Resources Bulletin 21, 645650.
Mckenzie, E. (1986) Autoregressive-moving-average processes with negative binomial and geometric marginal distributions. Adv. Appl. Prob. 18, 679705.
Mckenzie, E. (1988) The distributional structure of finite moving average processes. J. Appl. Prob. 25, 313321.
Steudel, H. J. and Wu, S. M. (1977) A time series approach to queueing systems with applications for modelling job-shop in-process inventories. Management Sci. 23, 745755.
Steudel, H. J., Pandit, S. M. and Wu, S. M. (1977) A multiple time series approach to modelling the manufacturing job-shop as a network of queues. Management Sci. 24, 456463.
Steutel, F. W. and Van Harn, K. (1979) Discrete analogues of self-decomposability and stability. Ann. Prob. 7, 893899.
Steutel, F. W. and Van Harn, K. (1986) Discrete operator self-decomposability and queueing networks. Commun. Statist.-Stochastic Models 2, 161169.
Steutel, F. W., Vervaat, W. and Wolfe, S. J. (1983) Integer valued branching processes with immigration. Adv. Appl. Prob. 15, 713725.
Stordahl, K. (1980) Analysis of telecommunications data by use of ARIMA models. In Forecasting Public Uitilities, ed. Anderson, O. D., North-Holland, Amsterdam, 8199.
Stordahl, K., Sollie, B. H. and Damsleth, E. (1979) Confidence limits for the expected telephone traffic in simulation models using ARMA-models. 9th International Teletraffic Congress, Malaga, Spain.
Teicher, H. (1954) On the multivariate Poisson distribution, Skandinavisk. Aktuarietidskrift 37, 19.
Terry, W. R. and Kumar, K. S. (1985) State vector time series analysis of queueing network simulation models. In Time Series Analysis: Theory and Practice 7, ed. Anderson, O. D., North-Holland, Amsterdam, 221229.
Weiss, G. (1975) Time reversibility of linear stochastic processes. J. Appl. Prob. 12, 831836.
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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
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