Extensions of the Generalized Pólya Process

Francisco Germán Badía (), Sophie Mercier () and Carmen Sangüesa ()
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Francisco Germán Badía: University of Zaragoza
Sophie Mercier: CNRS / Univ Pau & Pays Adour / E2S UPPA
Carmen Sangüesa: University of Zaragoza

Methodology and Computing in Applied Probability, 2019, vol. 21, issue 4, 1057-1085

Abstract: 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.

Keywords: Counting process; Non-homogeneous pure-birth process; Positive and negative dependence properties; Reliability theory; 60K10; 60E15 (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s11009-018-9663-y

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