Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-23T14:42:10.037Z Has data issue: false hasContentIssue false
  • English
  • Français

Autoregressive moving-average processes with negative-binomial and geometric marginal distributions

Published online by Cambridge University Press:  01 July 2016

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

Abstract

Some simple models are described which may be used for the modelling or generation of sequences of dependent discrete random variates with negative binomial and geometric univariate marginal distributions. The models are developed as analogues of well-known continuous variate models for gamma and negative exponential variates. The analogy arises naturally from a consideration of self-decomposability for discrete random variables. An alternative derivation is also given wherein both the continuous and the discrete variate processes arise simultaneously as measures on a process of overlapping intervals. The former is the process of interval lengths and the latter is a process of counts on these intervals.

Keywords

DISCRETE-VARIATE ARMA PROCESSESDISCRETE SELF-DECOMPOSABILITYINTERVAL PROCESSESBIVARIATE NEGATIVE BINOMIAL DISTRIBUTIONSTHINNING
Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 3 , September 1986 , pp. 679 - 705
Copyright
Copyright © Applied Probability Trust 1986 

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

Bondesson, L. (1979) On generalized gamma and generalized negative binomial convolutions. Part I. Scand. Act. J. , 125–146.Google Scholar
Cox, D. R. and Isham, V. (1980) Point Processes. Chapman and Hall, London.Google Scholar
Edwards, C. B. and Gurland, J. (1961) A class of distributions applicable to accidents. J. Amer. Statist. Assoc. 56, 5503–5517.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
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
Johnson, N. L. and Kotz, S. (1969) Discrete Distributions. Houghton-Mifflin, Boston.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.CrossRefGoogle Scholar
Lawrance, A. J. and Lewis, P. A. W. (1980) The exponential autoregressive-moving average EARMA (p, q) process. J. R. Statist. Soc. B 42, 150–161.Google Scholar
Lawrance, A. J. and Lewis, P. A. W. (1981) A new autoregressive time-series model in exponential variables (NEAR(1)). Adv. Appl. Prob. 13, 826–845.Google Scholar
Lewis, P. A. W. (1982) Simple multivariate time-series for simulations of complex systems. Proc. 1981 Winter Simulation Conf. , ed. Oren, T. I., Delfosse, C. M. and Shab, C. M., 389–390.Google Scholar
Loève, M. (1963) Probability Theory , 3rd edn. Van Nostrand, New York.Google Scholar
Patil, G. P., (Ed.) (1970) Random Counts in Scientific Work , Vol. I. The Pennsylvania State University Press, London.Google Scholar
Phatarfod, R. M. and Mardia, K. V. (1973) Some results for dams with Markovian inputs. J. Appl. Prob. 10, 166–180.CrossRefGoogle Scholar
Steutel, F. W. and Van Harn, K. (1979) Discrete analogues of self-decomposability and stability. Ann. Prob. 7, 893–899.CrossRefGoogle Scholar