Methodology and Computing in Applied Probability

, Volume 21, Issue 4, pp 1057–1085 | Cite as

Extensions of the Generalized Pólya Process

  • Francisco Germán Badía
  • Sophie MercierEmail author
  • Carmen Sangüesa


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)

60K10 60E15 


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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.


  1. Aalo V, Piboongungon T, Efthymoglou G (2005) Another look at the performance of MRC schemes in Nakagami-m fading channels with arbitrary parameters. IEEE Trans Commun 53(12):2002–2005CrossRefGoogle Scholar
  2. Asfaw ZG, Lindqvist BH (2015) Extending minimal repair models for repairable systems: a comparison of dynamic and heterogeneous extensions of a nonhomogeneous Poisson process. Reliab Eng Syst Saf 140:53–58CrossRefGoogle Scholar
  3. Asmussen S (2003) Applied probability and queues, Applications of Mathematics, vol 51, 2nd edn. Springer, New YorkGoogle Scholar
  4. Babykina G, Couallier V (2010) Advances in degradation modeling: applications to reliability, survival analysis, and finance. Birkhäuser Boston, Boston, pp 339–354. chap Modelling Recurrent Events for Repairable Systems Under Worse Than Old AssumptionCrossRefGoogle Scholar
  5. Badía F, Sangüesa C, Cha J (2018) Stochastic comparisons and ageing properties of generalized Pólya processes. J Appl Probab 55(1):233–253MathSciNetCrossRefGoogle Scholar
  6. Bedbur S, Kamps U (2017) Inference in a two-parameter generalized order statistics model. Statistics 51(5):1132–1142MathSciNetCrossRefGoogle Scholar
  7. Belzunce F, Lillo RE, Ruiz JM, Shaked M (2001) Stochastic comparisons of nonhomogeneous processes. Probab Eng Inf Sci 15:199–224MathSciNetCrossRefGoogle Scholar
  8. Belzunce F, Mercader JA, Ruiz JM (2003) Multivariate aging properties of epoch times of nonhomogeneous processes. J Multivar Anal 84:335–350MathSciNetCrossRefGoogle Scholar
  9. Billingsley P (1995) Probability and measure, 3rd edn. Wiley Series in Probability and statistics: probability and statistics. Wiley, New YorkGoogle Scholar
  10. Bordes L, Mercier S (2013) Extended geometric processes: semiparametric estimation and application to reliability. J Iran Stat Soc 12(1):1–34MathSciNetzbMATHGoogle Scholar
  11. Boyer M, Roux P (2016) A common framework embedding network calculus and event stream theory., working paper or preprint
  12. Cha JH (2014) Characterization of the generalized Pólya process and its applications. Adv Appl Probab 46:1148–1171MathSciNetCrossRefGoogle Scholar
  13. Cha J, Finkelstein M (2016) Justifying the Gompertz curve of mortality via the generalized Polya process of shocks. Theor Popul Biol 109:54–62CrossRefGoogle Scholar
  14. Cha JH, Finkelstein M (2017) New shock models based on the generalized Pólya process. Eur J Oper Res 251:1148–1171Google Scholar
  15. Chauvel C, Dauxois JY, Doyen L, Gaudoin O (2016) Parametric bootstrap goodness-of-fit tests for imperfect maintenance models. IEEE Trans Reliab 65(3):1343–1359CrossRefGoogle Scholar
  16. Cox D (1970) Renewal theory, 1st edn. Methuen & Co., LondonGoogle Scholar
  17. Daley DJ, Vere-Jones D (2003) An introduction to the theory of point processes - volume I: general theory and structure, 2nd edn. Springer Science & Business Media, New YorkzbMATHGoogle Scholar
  18. Doyen L, Gaudoin O (2004) Classes of imperfect repair models based on reduction of failure intensity or virtual age. Reliab Eng Syst Saf 84(1):45–56CrossRefGoogle Scholar
  19. Efthymoglou G, Aalo V (1995) Performance of RAKE receivers in Nakagami fading channel with arbitrary fading parameters. Electron Lett 31(18):1610–1612CrossRefGoogle Scholar
  20. Embrechts P, Hofert M (2013) A note on generalized inverses. Math Meth Oper Res 77(3):423–432MathSciNetCrossRefGoogle Scholar
  21. Gikhman II, Skorokhod AV (1969) Introduction to the theory of random processes. Translated from the Russian by Scripta Technica, Inc. Ont, PhiladelphiaGoogle Scholar
  22. Jacod J, Shiryaev AN (1987) Limit theorems for stochastic processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol 288. Springer, BerlinGoogle Scholar
  23. Janardan K (2005) Integral representation of a distribution associated with a pure birth process. Commun Stat: Theory Methods 34(11):2097–2103MathSciNetCrossRefGoogle Scholar
  24. Kamps U (1995a) A concept of generalized order statistics. J Stat Plan Infer 48 (11):1–23MathSciNetCrossRefGoogle Scholar
  25. Kamps U (1995b) A concept of generalized order statistics. Teubner Skripten zur Mathematischen Stochastik. [Teubner Texts on Mathematical Stochastics]. B. G. Teubner, StuttgartGoogle Scholar
  26. Kijima M (1989) Some results for repairable systems with general repair. J Appl Probab 26(1):89–102MathSciNetCrossRefGoogle Scholar
  27. Konno H (2010) On the exact solution of a generalized Pólya process. Adv Math Phys 2010, article ID 504267Google Scholar
  28. Lam Y (2007) The geometric process and its applications. World Scientific, HackensackCrossRefGoogle Scholar
  29. Landriault D, Willmot GE, Xu D (2014) On the analysis of time dependent claims in a class of birth process claim count models. Insurance Math Econom 58:168–173MathSciNetCrossRefGoogle Scholar
  30. Le Gat Y (2009) Une extension du processus de Yule pour la modélisation stochastique des événements récurrents. Application aux défaillances de canalisations d’eau sous pression. PhD thesis, Ecole Nationale du Génie Rural des Eaux et des Forêts (ENGREF), ParisGoogle Scholar
  31. Le Gat Y (2014) Extending the Yule process to model recurrent pipe failures in water supply networks. Urban Water J 11(8):617–630CrossRefGoogle Scholar
  32. Müller A, Stoyan D (2002) Comparison methods for stochastic models and risks. Wiley Series in Probability and Statistics. Wiley, ChichesterzbMATHGoogle Scholar
  33. Nadarajah S (2008) A review of results on sums of random variables. Acta Appl Math 103(2):131–140MathSciNetCrossRefGoogle Scholar
  34. Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer Series in Statistics. Springer, New YorkCrossRefGoogle Scholar
  35. Sheu SH, Chen YL, Chang CC, Zhang ZG (2016) A note on a two variable block replacement policy for a system subject to non-homogeneous pure birth shocks. Appl Math Model 40(5–6):3703–3712MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Francisco Germán Badía
    • 1
  • Sophie Mercier
    • 2
    Email author
  • Carmen Sangüesa
    • 3
  1. 1.Department of Statistical MethodsUniversity of ZaragozaZaragozaSpain
  2. 2.Laboratoire de Mathématiques et de leurs Applications / IPRA, UMR 5142CNRS / Univ Pau & Pays Adour / E2S UPPAPauFrance
  3. 3.Department of Statistical MethodsUniversity of ZaragozaZaragozaSpain

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