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Some Results and Applications of Geometric Counting Processes

Author

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  • Antonio Di Crescenzo

    (Università degli Studi di Salerno)

  • Franco Pellerey

    (Politecnico di Torino)

Abstract

Among Mixed Poisson processes, counting processes having geometrically distributed increments can be obtained when the mixing random intensity is exponentially distributed. Dealing with shock models and compound counting models whose shocks and claims occur according to such counting processes, we provide various comparison results and aging properties concerning total claim amounts and random lifetimes. Furthermore, the main characteristic distributions and properties of these processes are recalled and proved through a direct approach, as an alternative to those available in the literature. We also provide closed-form expressions for the first-crossing-time problem through monotone nonincreasing boundaries, and numerical estimates of first-crossing-time densities through other suitable boundaries. Finally, we present several applications in seismology, software reliability and other fields.

Suggested Citation

  • Antonio Di Crescenzo & Franco Pellerey, 2019. "Some Results and Applications of Geometric Counting Processes," Methodology and Computing in Applied Probability, Springer, vol. 21(1), pages 203-233, March.
    • Handle: RePEc:spr:metcap:v:21:y:2019:i:1:d:10.1007_s11009-018-9649-9
    DOI: 10.1007/s11009-018-9649-9
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    References listed on IDEAS

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    1. Cha, Ji Hwan & Finkelstein, Maxim, 2018. "On information-based residual lifetime in survival models with delayed failures," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 209-216.
    2. Belzunce, Felix & Ortega, Eva-Maria & Pellerey, Franco & Ruiz, Jose M., 2006. "Variability of total claim amounts under dependence between claims severity and number of events," Insurance: Mathematics and Economics, Elsevier, vol. 38(3), pages 460-468, June.
    3. Joag-dev, Kumar & Kochar, Subhash & Proschan, Frank, 1995. "A general composition theorem and its applications to certain partial orderings of distributions," Statistics & Probability Letters, Elsevier, vol. 22(2), pages 111-119, February.
    4. M. Sreehari & R. Vasudeva, 2012. "Characterizations of multivariate geometric distributions in terms of conditional distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 75(2), pages 271-286, February.
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    Cited by:

    1. Abdolsaeed Toomaj & Antonio Di Crescenzo, 2020. "Connections between Weighted Generalized Cumulative Residual Entropy and Variance," Mathematics, MDPI, vol. 8(7), pages 1-27, July.
    2. Dheeraj Goyal & Nil Kamal Hazra & Maxim Finkelstein, 2022. "On Properties of the Phase-type Mixed Poisson Process and its Applications to Reliability Shock Modeling," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2933-2960, December.

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