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Extensions of the Generalized Pólya Process

Author

Listed:
  • Francisco Germán Badía

    (University of Zaragoza)

  • Sophie Mercier

    (CNRS / Univ Pau & Pays Adour / E2S UPPA)

  • Carmen Sangüesa

    (University of Zaragoza)

Abstract

A new self-exciting counting process is here considered, which extends the generalized Pólya process introduced by Cha (Adv Appl Probab 46:1148–1171, 2014). Contrary to Cha’s original model, where the intensity of the process (linearly) increases at each jump time, the extended version allows for more flexibility in the dependence between the point-wise intensity of the process at some time t and the number of already observed jumps. This allows the “extended Pólya process” to be appropriate, e.g., for describing successive failures of a system subject to imperfect but effective repairs, where the repair can lower the intensity of the underlying counting process. Probabilistic properties of the new process are studied (construction from a homogeneous pure-birth process, conditions of non explosion, computation of distributions, convergence of a sequence of such processes, ...) and its connection with Generalized Order Statistics is highlighted. Positive dependence properties are next explored. Finally, the maximum likelihood method is considered in a parametric setting and tested on a few simulated data sets, to highlight the practical use of the new process in an application context.

Suggested Citation

  • Francisco Germán Badía & Sophie Mercier & Carmen Sangüesa, 2019. "Extensions of the Generalized Pólya Process," Methodology and Computing in Applied Probability, Springer, vol. 21(4), pages 1057-1085, December.
    • Handle: RePEc:spr:metcap:v:21:y:2019:i:4:d:10.1007_s11009-018-9663-y
    DOI: 10.1007/s11009-018-9663-y
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    References listed on IDEAS

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    1. Cha, Ji Hwan & Finkelstein, Maxim, 2016. "Justifying the Gompertz curve of mortality via the generalized Polya process of shocks," Theoretical Population Biology, Elsevier, vol. 109(C), pages 54-62.
    2. Belzunce, Félix & Mercader, José A. & Ruiz, José M., 2003. "Multivariate aging properties of epoch times of nonhomogeneous processes," Journal of Multivariate Analysis, Elsevier, vol. 84(2), pages 335-350, February.
    3. Paul Embrechts & Marius Hofert, 2013. "A note on generalized inverses," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(3), pages 423-432, June.
    4. Landriault, David & Willmot, Gordon E. & Xu, Di, 2014. "On the analysis of time dependent claims in a class of birth process claim count models," Insurance: Mathematics and Economics, Elsevier, vol. 58(C), pages 168-173.
    5. Asfaw, Zeytu Gashaw & Lindqvist, Bo Henry, 2015. "Extending minimal repair models for repairable systems: A comparison of dynamic and heterogeneous extensions of a nonhomogeneous Poisson process," Reliability Engineering and System Safety, Elsevier, vol. 140(C), pages 53-58.
    6. Konanur Janardan, 2005. "A discrete distribution associated with a pure birth process," Statistical Papers, Springer, vol. 46(4), pages 587-597, October.
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    Cited by:

    1. Kerry Fendick & Ward Whitt, 2022. "Heavy traffic limits for queues with non-stationary path-dependent arrival processes," Queueing Systems: Theory and Applications, Springer, vol. 101(1), pages 113-135, June.

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