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Stationary Time Series Models with Exponential Dispersion Model Margins

Part of: Stochastic processes Inference from stochastic processes

Published online by Cambridge University Press:  14 July 2016

Bent Jørgensen  and
Peter Xue-Kun Song
Bent Jørgensen*
Affiliation:
University of British Columbia
Peter Xue-Kun Song*
Affiliation:
York University
*
∗Postal address: Department of Statistics, University of British Columbia, 333–6356 Agricultural Road, Vancouver B.C., Canada V6T 1Z2
∗∗Postal address: Department of Mathematics and Statistics, York University, 4700 Keele Street, North York, Ontario, Canada M3J 1P3.
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Abstract

We consider a class of stationary infinite-order moving average processes with margins in the class of infinitely divisible exponential dispersion models. The processes are constructed by means of the thinning operation of Joe (1996), generalizing the binomial thinning used by McKenzie (1986, 1988) and Al-Osh and Alzaid (1987) for integer-valued time series. As a special case we obtain a class of autoregressive moving average processes that are different from the ARMA models proposed by Joe (1996). The range of possible marginal distributions for the new models is extensive and includes all infinitely divisible distributions with finite moment generating functions, hereunder many known discrete, continuous and mixed distributions.

Keywords

Autoregressive processconvolution-closed familyinfinite divisibilitymoving average process of infinite ordernon-normal time seriessaddlepoint approximationsmall-dispersion asymptoticsthinning

MSC classification

Primary: 60G10: Stationary processes
Secondary: 62M10: Time series, auto-correlation, regression, etc.
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 1 , March 1998 , pp. 78 - 92
Copyright
Copyright © Applied Probability Trust 1998 

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Footnotes

Research supported by the Natural Sciences and Engineering Research Council of Canada.

References

Al-Osh, M. A., and Aly, E. A. A. (1992). First-order autoregressive time series with negative binomial and geometric marginals. Commun. Statist. A 21, 2483–2492.Google Scholar
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
Alzaid, A. A., and Al-Osh, M. A. (1993). Some autoregressive moving average processes with generalized Poisson marginal distributions. Ann. Inst. Statist. Math. 45, 223–232.Google Scholar
Barndorff-Nielsen, O. E. and Jörgensen, B. (1991). Some parametric models on the simplex. J. Multivar. Anal. 39, 106–116.Google Scholar
Brockwell, P. J., and Davis, R. A. (1987). Time Series: Theory and Methods. Springer, New York.Google Scholar
Chung, K. L. (1974). Course in Probability Theory. 2nd edn. Academic Press, New York.Google Scholar
Consul, P. C. (1989). Generalized Poisson Distribution: Properties and Applications. Marcel Dekker, New York.Google Scholar
Dimri, V. (1992). Deconvolution and Inverse Theory. Elsevier, Amsterdam.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and its Applications. Vol. 2, 2nd edn. Wiley, New York.Google 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
Jacobs, P. A., and Lewis, P. A. W. (1977). A mixed autoregressive-moving average exponential sequence and point process (EARMA 1,1). Adv. Appl. Prob. 9, 87–104.Google Scholar
Jansson, P. A. (1984). Deconvolution with Applications in Spectroscopy. Academic Press, New York.Google Scholar
Joe, H. (1996). Time series models with univariate margins in the convolution-closed infinitely divisible class. J. Appl. Prob. 33, 664–677.Google Scholar
Jörgensen, B. (1986). Some properties of exponential dispersion models. Scand. J. Statist. 13, 187–198.Google Scholar
Jörgensen, B. (1987). Exponential dispersion models (with discussion). J. Roy. Statist. Soc. B 49, 127–162.Google Scholar
Jörgensen, B. (1992). Exponential dispersion models and extensions: a review. Int. Statist. Rev. 60, 5–20.Google Scholar
Jörgensen, B., and Souza, M. P. (1994). Fitting Tweedie's compound Poisson model to insurance claims data. Scand. Act. J. 69–93.Google Scholar
Jörgensen, B., Martínez, J. R., and Tsao, M. (1994). Asymptotic behaviour of the variance function. Scand. J. Statist. 21, 223–243.Google Scholar
Jörgensen, B. and Martínez, J. R. (1997). Tauber theory for infinitely divisible variance functions. Bernoulli 3, 213–224.Google Scholar
Lawrance, A. J. (1982). The innovation distribution of a gamma distributed autoregressive process. Scand. J. Statist. 9, 234–236.Google Scholar
Lawrance, A. J., and Lewis, P. A. W. (1977). An exponential moving average sequence and point process (EMA 1). J. Appl. Prob. 14, 98–113.Google Scholar
Letac, G., and Mora, M. (1990). Natural real exponential families with cubic variances. Ann. Statist. 18, 1–37.Google Scholar
Lewis, P. A. W. (1983). Generating negatively correlated gamma variates using the beta-gamma transform. In Proc. 1983 Winter Simulation Conf. ed. Roberts, S., Banks, J. and Schmeiser, B. IEEE Press, New York. pp. 175–176.Google Scholar
Lewis, P. A. W., McKenzie, E., and Hugus, D. K. (1989). Gamma processes. Commun. Statist.–Stoch. Models 5, 1–30.Google Scholar
McCormick, W. P., and Park, Y. S. (1992). Asymptotic analysis of extremes from autoregressive negative binomial processes. J. Appl. Prob. 29, 904–920.Google Scholar
McKenzie, E. (1986). Autoregressive moving average processes with negative-binomial and geometric marginal distributions. Adv. Appl. Prob. 18, 679–705.Google Scholar
McKenzie, E. (1988). Some ARMA models for dependent sequences of Poisson counts. Adv. Appl. Prob. 20, 822–835.Google Scholar
Morris, C. N. (1982). Natural exponential families with quadratic variance functions. Ann. Statist. 10, 65–80.Google Scholar
Seshadri, V. (1992). General exponential models on the unit simplex and related multivariate inverse Gaussian distributions. Statist. Prob. Lett. 14, 385–391.Google Scholar
Yanagimoto, T. (1989). The inverse binomial distribution as a statistical model. Commun. Statist. 18, 3625–3633.Google Scholar