Abstract
Some simple models are introduced which may be used for modelling or generating sequences of dependent discrete random variables with generalized Poisson marginal distribution. Our approach for building these models is similar to that of the Poisson ARMA processes considered by Al-Osh and Alzaid (1987,J. Time Ser. Anal.,8, 261–275; 1988,Statist. Hefte,29, 281–300) and McKenzie (1988,Adv. in Appl. Probab.,20, 822–835). The models have the same autocorrelation structure as their counterparts of standard ARMA models. Various properties, such as joint distribution, time reversibility and regression behavior, for each model are investigated.
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Al-Osh, M. A. and Alzaid, A. A. (1987). First-order integer-valued autoregressive (INAR(1)) process,J. Time Ser. Anal.,8, 261–275.
Al-Osh, M. A. and Alzaid, A. A. (1988). Integer-valued moving average (INMA) process,Statist. Hefte,29, 281–300.
Alzaid, A. A. and Al-Osh, M. A. (1988). First-order integer-valued autoregressive (INAR(1)) process: distributional and regression properties,Statist. Neerlandica,42, 53–61.
Alzaid, A. A. and Al-Osh, M. A. (1990). Integer-valuedpth order autoregressive structure (INAR(p)) process,J. Appl. Probab.,27, 314–324.
Consul, P. C. (1975). Some new characterizations of discrete Lagrangian distributions,Statistical Distributions in Scientific Work, Vol. 3 (eds. G. P. Patil, S. Kotz and J. Ord), 279–290, Reidel, Dordrecht.
Consul, P. C. (1989).Generalized Poisson Distribution: Properties and Applications, Marcel Dekker, New York.
Consul, P. C. and Jain, G. C. (1973). A generalization of Poisson distribution,Technometrics,15, 791–799.
Consul, P. C. and Mittal, S. P. (1975). A new urn model with predetermined strategy,Biometrical J.,17, 67–75.
Consul, P. C. and Mittal, S. P. (1977). Some discrete multinomial probability models with predetermined strategy,Biometrical J.,19, 167–173.
Consul, P. C. and Shenton, L. R. (1973). Some interesting properties of Lagrange distributions,Comm. Statist.,2, 263–272.
Janardan, K. G., Kerster, H. W. and Schaeffer, D. J. (1979). Biological applications of the Lagrangian Poisson distribution,Bioscience,29, 599–602.
Kumar, A. (1981). Some application of Lagrangian distributions in queueing theory and epidemiology,Comm. Statist. Theory Methods,10, 1429–1436.
McKenzie, E. (1985). Some simple models for discrete variate time series,Water Resources Bulletin,21, 645–650.
McKenzie, E. (1988). Some ARMA models for dependent sequences of Poisson counts,Adv. in Appl. Probab.,20, 822–835.
Shenton, L. R. (1986). Quasibinomial distributions,Encyclopedia of Statistical Sciences (eds. S. Kotz and N. L. Johnson), Vol. 7, 458–460.
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Alzaid, A.A., Al-Osh, M.A. Some autoregressive moving average processes with generalized Poisson marginal distributions. Ann Inst Stat Math 45, 223–232 (1993). https://doi.org/10.1007/BF00775809
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DOI: https://doi.org/10.1007/BF00775809