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A general multivariate lifetime model with a multivariate additive process as conditional hazard rate increment process

Author

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  • Sophie Mercier

    (Universite de Pau et des Pays de l’Adour)

  • Carmen Sangüesa

    (University of Zaragoza)

Abstract

The object of the present paper is the study of the joint lifetime of d components subject to a common stressful external environment. Out of the stressing environment, the components are independent and the lifetime of each component is characterized by its failure (hazard) rate function. The impact of the external environment is modelled through an increase in the individual failure rates of the components. The failure rate increments due to the environment increase over time and they are dependent among components. The evolution of the joint failure rate increments is modelled by a non negative multivariate additive process, which include Lévy processes and non-homogeneous compound Poisson processes, hence encompassing several models from the previous literature. A full form expression is provided for the multivariate survival function with respect to the intensity measure of a general additive process, using the construction of an additive process from a Poisson random measure (or Poisson point process). The results are next specialized to Lévy processes and other additive processes (time-scaled Lévy processes, extended Lévy processes and shock models), thus providing simple and easily computable expressions. All results are provided under the assumption that the additive process has bounded variations, but it is possible to relax this assumption by means of approximation procedures, as is shown for the last model of this paper.

Suggested Citation

  • Sophie Mercier & Carmen Sangüesa, 2023. "A general multivariate lifetime model with a multivariate additive process as conditional hazard rate increment process," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 86(1), pages 91-129, January.
    • Handle: RePEc:spr:metrik:v:86:y:2023:i:1:d:10.1007_s00184-022-00864-3
    DOI: 10.1007/s00184-022-00864-3
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    References listed on IDEAS

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    1. Guida, M. & Postiglione, F. & Pulcini, G., 2012. "A time-discrete extended gamma process for time-dependent degradation phenomena," Reliability Engineering and System Safety, Elsevier, vol. 105(C), pages 73-79.
    2. Sophie Mercier & Hai Ha Pham, 2016. "A Random Shock Model with Mixed Effect, Including Competing Soft and Sudden Failures, and Dependence," Methodology and Computing in Applied Probability, Springer, vol. 18(2), pages 377-400, June.
    3. Y. Kebir, 1991. "On hazard rate processes," Naval Research Logistics (NRL), John Wiley & Sons, vol. 38(6), pages 865-876, December.
    4. Kallsen, Jan & Tankov, Peter, 2006. "Characterization of dependence of multidimensional Lévy processes using Lévy copulas," Journal of Multivariate Analysis, Elsevier, vol. 97(7), pages 1551-1572, August.
    5. Mercier, Sophie & Pham, Hai Ha, 2017. "A bivariate failure time model with random shocks and mixed effects," Journal of Multivariate Analysis, Elsevier, vol. 153(C), pages 33-51.
    6. Zeina Al Masry & Sophie Mercier & Ghislain Verdier, 2017. "Approximate Simulation Techniques and Distribution of an Extended Gamma Process," Methodology and Computing in Applied Probability, Springer, vol. 19(1), pages 213-235, March.
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