Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-23T20:18:20.357Z Has data issue: false hasContentIssue false
  • English
  • Français

First-order autoregressive gamma sequences and point processes

Published online by Cambridge University Press:  01 July 2016

D. P. Gaver  and
P. A. W. Lewis
D. P. Gaver
Affiliation:
Naval Postgraduate School, Monterey
P. A. W. Lewis
Affiliation:
Naval Postgraduate School, Monterey
Get access Rights & Permissions [Opens in a new window]

Abstract

It is shown that there is an innovation process {∊n} such that the sequence of random variables {Xn} generated by the linear, additive first-order autoregressive scheme Xn = pXn-1 + ∊n are marginally distributed as gamma (λ, k) variables if 0 ≦p ≦ 1. This first-order autoregressive gamma sequence is useful for modelling a wide range of observed phenomena. Properties of sums of random variables from this process are studied, as well as Laplace-Stieltjes transforms of adjacent variables and joint moments of variables with different separations. The process is not time-reversible and has a zero-defect which makes parameter estimation straightforward. Other positive-valued variables generated by the first-order autoregressive scheme are studied, as well as extensions of the scheme for generating sequences with given marginal distributions and negative serial correlations.

Keywords

AUTOREGRESSIVE PROCESSESEXPONENTIAL DISTRIBUTIONSGAMMA DISTRIBUTIONSSERIAL CORRELATIONSMIXED EXPONENTIAL DISTRIBUTIONSZERO-DEFECTINNOVATION PROCESSEAR(1) PROCESSGAR(1) PROCESSMEAR(1) PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 3 , September 1980 , pp. 727 - 745
Copyright
Copyright © Applied Probability Trust 1980 

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

Bernier, J. (1970) Inventaire des modèles et processus stochastique applicables de la description des déluts journaliers des riviers. Rev. Inst. Internat. Statist. 38, 50–71.CrossRefGoogle Scholar
Chernick, M. R. (1977) A limit theorem for the maximum of an exponential autoregressive process. Tech. Report No. 14, SIMS, Dept. of Statistics, Stanford University.Google Scholar
Cox, D. R. (1955) Some statistical methods connected with series of events. J. R. Statist. Soc. B 17, 129–164.Google Scholar
Cox, D. R. (1962) Renewal Theory. Methuen, London.Google Scholar
Cox, D. R. and Lewis, P. A. W. (1966) The Statistical Analysis of Series of Events. Methuen, London.CrossRefGoogle Scholar
Downton, F. (1970) Bivariate exponential distributions of reliability theory. J. R. Statist. Soc. B 32, 408–417.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications. Wiley, New York.Google Scholar
Gaver, D. P. (1972) Point process problems in reliability. In Stochastic Point Processes, ed Lewis, P. A. W., Wiley, New York, 775–800.Google Scholar
Hammersley, J. and Handscomb, D. C. (1964) Monte Carlo Methods, Wiley, New York.CrossRefGoogle Scholar
Hawkes, A. G. (1972) Mutually exciting point processes. In Stochastic Point Processes, ed. Lewis, P. A. W., Wiley, New York, 261–271.Google Scholar
Jacobs, P. A. (1978) A closed cyclic queueing network with dependent exponential service times. J. Appl. Prob. 15, 573–589.CrossRefGoogle 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.CrossRefGoogle Scholar
Lampard, D. G. A. (1968) A stochastic process whose intervals between events form a first order Markov chain. I. J. Appl. Prob. 5, 648–668.CrossRefGoogle Scholar
Lawrance, A. J. (1972) Some models for series of univariate events. In Stochastic Point Processes, ed. Lewis, P. A. W., Wiley, New York, 199–256.Google Scholar
Lawrance, A. J. (1978) Some autoregressive models for point processes. In Proc. Bolyai Mathematical Society Colloquium on Point Processes and Queueing Theory, North-Holland, Amsterdam.Google Scholar
Lawrance, A. J. (1980) The mixed exponential solution to the first-order autoregressive model. J. Appl. Prob. 17, 546–552.CrossRefGoogle Scholar
Lawrance, A. J. and Lewis, P. A. W. (1977) A moving average exponential point process (EMA1). J. Appl. Prob. 14, 98–113.CrossRefGoogle Scholar
Lawrance, A. J. and Lewis, P. A. W. (1980) The exponential autoregressive-moving average process EARMA(p, q) . J. R. Statist. Soc. B. CrossRefGoogle Scholar
Lewis, P. A. W. (1978) Simple models for positive-valued and discrete-valued time series with arma correlation structure. In Multivariate Analysis V., ed. Krishnaiah, P. R., North-Holland, Amsterdam, 151–166.Google Scholar
Lewis, P. A. W. and Shedler, G. S. (1978) Analysis and modelling of point processes in computer systems. Bull. ISI XLVII (2), 193–210.Google Scholar
Linhart, H. (1970) The gamma point process. S. Afr. Statist. J. 4, 1–17.Google Scholar
Linhart, H. and Zucchini, W. (1972) Parametric models for point processes. S. Afr. Statist. J. 6, 19–26.Google Scholar
Mallows, C. L. (1967) Linear processes are nearly Gaussian. J. Appl Prob. 4, 313–329.CrossRefGoogle Scholar
Marshall, A. W. and Olkin, I. (1967) A multivariate exponential distribution. J. Amer. Statist. Assoc. 62, 30–44.Google Scholar
Moran, P. A. P. (1967) Testing for correlation between non-negative variates. Biometrika 54, 385–394.CrossRefGoogle ScholarPubMed
Rosenblatt, M. (1971) Markov Processes, Structure and Asymptotic Behaviour. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Shanbhag, D. N., Pestana, D. and Sreehari, H. (1977) Some further results in infinite divisibility. Math. Proc. Camb. Phil. Soc. 82, 289–295.CrossRefGoogle Scholar
Shanbhag, D. N. and Sreehari, M. (1977) On certain self-decomposable distributions. Z. Wahrscheinlichkeitsth 38, 217–222.Google Scholar
Thorin, O. (1977a) On the infinite divisibility of the Pareto distribution. Scand. Actuarial J. 4, 31–40.CrossRefGoogle Scholar
Thorin, O. (1977b) On the infinite divisibility of the log normal distribution. Scand. Actuarial J. 4, 121–148.Google Scholar
Wold, H. (1948) Sur les processus stationnaires ponctuels. Colloques Internat. CNRS 13, 75–86.Google Scholar