Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-11T10:51:53.961Z Has data issue: false hasContentIssue false
  • English
  • Français

Some ARMA models for dependent sequences of poisson counts

Published online by Cambridge University Press:  01 July 2016

Ed Mckenzie
Ed Mckenzie*
Affiliation:
University of Strathclyde
*
∗Postal address: Department of Mathematics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

BINOMIAL THINNINGDISCRETE SELF-DECOMPOSABILITYPOISSON ARMA PROCESSESMULTIVARIATE POISSON DISTRIBUTIONTIME-REVERSIBILITYMULTINOMIAL THINNINGVECTOR AR(1) PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 4 , December 1988 , pp. 822 - 835
Copyright
Copyright © Applied Probability Trust 1988 

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

Al-Osh, M. A. and Alzaid, A. A. (1987) First order integer-valued autoregressive (INAR(1)) process. J. Time Series Anal. 8, 261–275.CrossRefGoogle Scholar
Box, G. E. P. and Jenkins, G. M. (1970) Time series Analysis: Forecasting and Control. Holden-Day, San Francisco.Google Scholar
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, 153–188.Google Scholar
Chiang, C. L. (1980) An Introduction to Stochastic Processes and their Applications. Krieger, Huntington, NY.Google Scholar
Dwass, M. and Teicher, H. (1957) On infinitely divisible random vectors. Ann. Math. Statist. 28, 461–470.CrossRefGoogle Scholar
Gaver, D. P. and Lewis, P. A. W. (1980) First order autoregressive gamma sequences and point processes. Adv. Appl. Prob. 12, 727–745.Google Scholar
Griffiths, R. C., Milne, R. K. and Wood, R. (1979) Aspects of correlation in bivariate Poisson distributions and processes. Austral. J. Statist. 21, 238–255.Google Scholar
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, 94–105.Google Scholar
Jacobs, P. A. and Lewis, P. A. W. (1978b) Discrete time series generated by mixtures II: Asymptotic properties. J. R. Statist. Soc. B 40, 222–228.Google Scholar
Jacobs, P. A. and Lewis, P. A. W. (1983) Stationary discrete autoregressive moving average time series generated by mixtures J. Time Series Anal. 4, 19–36.Google Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Lawrance, A. J. and Lewis, P. A. W. (1980) The exponential autoregressive moving average EARMA(p, q) process. J. R. Statist. Soc. B42, 150–161.Google Scholar
Mckendrick, A. G. (1962) The application of mathematics to medical problems. Proc. Edinburgh. Math. Soc. XLIV, 98–130.Google Scholar
Mckenzie, E. (1985) Some simple models for discrete variate time series. Water Resources Bulletin 21, 645–650.Google Scholar
Mckenzie, E. (1986) Autoregressive-moving-average processes with negative binomial and geometric marginal distributions. Adv. Appl. Prob. 18, 679–705.CrossRefGoogle Scholar
Mckenzie, E. (1988) The distributional structure of finite moving average processes. J. Appl. Prob. 25, 313–321.Google Scholar
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, 745–755.CrossRefGoogle Scholar
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, 456–463.Google Scholar
Steutel, F. W. and Van Harn, K. (1979) Discrete analogues of self-decomposability and stability. Ann. Prob. 7, 893–899.CrossRefGoogle Scholar
Steutel, F. W. and Van Harn, K. (1986) Discrete operator self-decomposability and queueing networks. Commun. Statist.-Stochastic Models 2, 161–169.Google Scholar
Steutel, F. W., Vervaat, W. and Wolfe, S. J. (1983) Integer valued branching processes with immigration. Adv. Appl. Prob. 15, 713–725.Google Scholar
Stordahl, K. (1980) Analysis of telecommunications data by use of ARIMA models. In Forecasting Public Uitilities, ed. Anderson, O. D., North-Holland, Amsterdam, 81–99.Google Scholar
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.Google Scholar
Teicher, H. (1954) On the multivariate Poisson distribution, Skandinavisk. Aktuarietidskrift 37, 1–9.Google Scholar
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, 221–229.Google Scholar
Weiss, G. (1975) Time reversibility of linear stochastic processes. J. Appl. Prob. 12, 831–836.Google Scholar