Hostname: page-component-6898cfcb55-kwdml Total loading time: 0 Render date: 2026-11-11T14:24:12.150Z Has data issue: false hasContentIssue false
  • English
  • Français

Time series models with univariate margins in the convolution-closed infinitely divisible class

Part of:Stochastic processesInference from stochastic processes

Published online by Cambridge University Press:  14 July 2016

Harry Joe
Harry Joe*
Affiliation:
University of British Columbia
*
∗ Postal address: Department of Statistics, The University of British Columbia, 6356 Agricultural Road, Vancouver, British Columbia, Canada V6T 1Z2.
Get accessRights & Permissions [Opens in a new window]

Abstract

A unified way of obtaining stationary time series models with the univariate margins in the convolution-closed infinitely divisible class is presented. Special cases include gamma, inverse Gaussian, Poisson, negative binomial, and generalized Poisson margins. ARMA time series models obtain in the special case of normal margins, sometimes in a different stochastic representation. For the gamma and Poisson margins, some previously defined time series models are included, but for the negative binomial margin, the time series models are different and, in several ways, better than previously defined time series models. The models are related to multivariate distributions that extend a univariate distribution in the convolution-closed infinitely divisible class. Extensions to the non-stationary case and possible applications to modelling longitudinal data are mentioned.

Keywords

NON-NORMAL TIME SERIESINFINITELY DIVISIBLECOUNT DATANEGATIVE BINOMIAL

MSC classification

Primary:60G10: Stationary processes
Secondary:62M10: Time series, auto-correlation, regression, etc.

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 3 , September 1996, pp. 664 - 677
Copyright
Copyright © Applied Probability Trust 1996 

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.)

Article purchase

Temporarily unavailable

References

Al-Osh, M. A. and Aly, E. A. A. (1992) First order autoregressive time series with negative binomial and geometric marginals. Commun. Statist. A21, 2483–2492.CrossRefGoogle 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
Al-Osh, M. A. and Alzaid, A. A. (1991) Binomial autoregressive moving average models. Commun. Statist. Stoch. Models 7, 261–282.CrossRefGoogle Scholar
Al-Osh, M. A. and Alzaid, A. A. (1993) Some gamma processes based on the Dirichlet-gamma transformation. Commun. Statist. Stoch. Models 9, 123–143.CrossRefGoogle Scholar
Al-Osh, M. A. and Alzaid, A. A. (1994) A class of non-negative time series processes with ARMA structure. J. Appl. Statist. Sci. 1, 313–322.Google Scholar
Alzaid, A. A. and Al-Osh, M. A. (1991) An integer-valued pth order autoregressive structure (INAR(p)) process. J. Appl. Prob. 27, 314–324.CrossRefGoogle Scholar
Alzaid, A. A. and Al-Osh, M. A. (1993) Some autoregressive moving average processes with generalized Poisson marginal distributions. Ann. Int. Statist. Math. 45, 223–232.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. and Jorgensen, B. (1991) Some parametric models on the simplex. J. Multivariate Anal. 39, 106–116.CrossRefGoogle Scholar
Consul, P. C. (1989) Generalized Poisson Distribution: Properties and Applications. Marcel Dekker, New York.Google Scholar
Jorgensen, B. (1986) Some properties of exponential dispersion models. Scand. J. Statist. 13, 187–197.Google Scholar
Jorgensen, B. (1992) Exponential dispersion models and extensions: a review. Int. Statist. Rev. 60, 5–20.CrossRefGoogle Scholar
Lawless, J. (1987) Negative binomial and mixed Poisson regression. Canad. J. Statist. 15, 209–225.CrossRefGoogle 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.CrossRefGoogle Scholar
Marshall, A. W. and Olkin, I. (1985) A family of bivariate distributions generated by the bivariate Bernoulli distribution. J. Amer. Statist. Assoc. 80, 332–338.CrossRefGoogle Scholar
Mccormick, W. P. and Park, Y. S. (1992) Asymptotic analysis of extremes from autoregressive negative binomial processes. J. Appl. Prob. 29, 904–920.CrossRefGoogle Scholar
Mckenzie, E. (1985) Some simple models for discrete variate time series. Water Resources Bull. 21, 645–650.CrossRefGoogle 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) Some ARMA models for dependent sequences of Poisson counts. Adv. Appl. Prob. 20, 822–835.CrossRefGoogle Scholar
Olkin, E, Petkau, A. J. and Zidek, J. V. (1981) A comparison of n estimators for the binomial distribution. J. Amer. Statist. Assoc. 76, 637–642.Google Scholar
Teicher, H. (1954) On the multivariate Poisson distribution. Skand. Akt. 37, 1–9.Google Scholar
Zeger, S. L. and Liang, K.-Y. (1986) Longitudinal data analysis for discrete and continuous outcomes. Biometrics 42, 121–130.CrossRefGoogle ScholarPubMed