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On the Mixed Extended Generalized Pólya Process and Its Stochastic Intensity Paradox

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  • Ji Hwan Cha

    (Ewha Womans University)

  • Maxim Finkelstein

    (University of the Free State
    University of Strathclyde)

Abstract

A new class of counting processes generated by the mixture of the extended generalized Pólya process is defined and its properties are studied. The general form of the corresponding stochastic intensity is derived. Specifying geometric, negative binomial, Poisson, and binomial distributions as the mixing distributions, four parametric classes of counting processes are defined and stochastically characterized. It is shown that relevant monotonicity properties of the corresponding stochastic intensities do not follow ‘direct intuition’ and can dramatically change depending on the mixing distribution. The practical meaning of the considered parametric models is also interpreted from the reliability point of view.

Suggested Citation

  • Ji Hwan Cha & Maxim Finkelstein, 2025. "On the Mixed Extended Generalized Pólya Process and Its Stochastic Intensity Paradox," Methodology and Computing in Applied Probability, Springer, vol. 27(3), pages 1-18, September.
    • Handle: RePEc:spr:metcap:v:27:y:2025:i:3:d:10.1007_s11009-025-10195-1
    DOI: 10.1007/s11009-025-10195-1
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    References listed on IDEAS

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    1. Navarro, Jorge & Fernández-Martínez, Pedro, 2021. "Redundancy in systems with heterogeneous dependent components," European Journal of Operational Research, Elsevier, vol. 290(2), pages 766-778.
    2. Maxim Finkelstein, 2008. "Failure Rate Modelling for Reliability and Risk," Springer Series in Reliability Engineering, Springer, number 978-1-84800-986-8, March.
    3. Alan G. Hawkes, 2018. "Hawkes processes and their applications to finance: a review," Quantitative Finance, Taylor & Francis Journals, vol. 18(2), pages 193-198, February.
    4. Isham, Valerie & Westcott, Mark, 1979. "A self-correcting point process," Stochastic Processes and their Applications, Elsevier, vol. 8(3), pages 335-347, May.
    5. Francisco Germán Badía & María Dolores Berrade, 2022. "On the Residual Lifetime and Inactivity Time in Mixtures," Mathematics, MDPI, vol. 10(15), pages 1-20, August.
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