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The max-INAR(1) model for count processes

Author

Listed:
  • Manuel G. Scotto

    (Universidade de Lisboa)

  • Christian H. Weiß

    (Helmut Schmidt University)

  • Tobias A. Möller

    (Helmut Schmidt University)

  • Sónia Gouveia

    (Universidade de Aveiro)

Abstract

This paper proposes a discrete counterpart of the conventional max-autoregressive process of order one. It is based on the so-called binomial thinning operator and driven by a sequence of independent and identically distributed nonnegative integer-valued random variables with either regularly varying right tail or exponential-type right tail. Basic probabilistic and statistical properties of the process are discussed in detail, including the analysis of conditional moments, transition probabilities, the existence and uniqueness of a stationary distribution, and the relationship between the observations’ and innovations’ distribution. We also provide conditions on the marginal distribution of the process to ensure that the innovations’ distribution exists and is well defined. Several examples of families of distributions satisfying such conditions are presented, but also some counterexamples are analyzed. Furthermore, the analysis of its extremal behavior is also considered. In particular, we look at the limiting distribution of sample maxima and its corresponding extremal index.

Suggested Citation

  • Manuel G. Scotto & Christian H. Weiß & Tobias A. Möller & Sónia Gouveia, 2018. "The max-INAR(1) model for count processes," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(4), pages 850-870, December.
    • Handle: RePEc:spr:testjl:v:27:y:2018:i:4:d:10.1007_s11749-017-0573-z
    DOI: 10.1007/s11749-017-0573-z
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    References listed on IDEAS

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    1. Schweer, Sebastian & Weiß, Christian H., 2014. "Compound Poisson INAR(1) processes: Stochastic properties and testing for overdispersion," Computational Statistics & Data Analysis, Elsevier, vol. 77(C), pages 267-284.
    2. Marta Ferreira & Helena Ferreira, 2013. "Extremes of multivariate ARMAX processes," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(4), pages 606-627, November.
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    Cited by:

    1. Paolo Gorgi, 2020. "Beta–negative binomial auto‐regressions for modelling integer‐valued time series with extreme observations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(5), pages 1325-1347, December.
    2. Aknouche, Abdelhakim & Scotto, Manuel, 2022. "A multiplicative thinning-based integer-valued GARCH model," MPRA Paper 112475, University Library of Munich, Germany.

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