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.
Counting process Non-homogeneous pure-birth process Positive and negative dependence properties Reliability theory
Mathematics Subject Classification (2010)
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The authors thank the Editor and the two Reviewers for their careful reading and constructive remarks, which allowed them to better the readability of the paper and correct some imprecision in the paper.
This work has been supported by the Spanish government research project MTM2015-63978(MINECO/FEDER). The first and third authors acknowledge the support of DGA S11 and E64, respectively.
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