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

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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.

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References

  • Abdel-Hameed M (2014) Lévy processes and their applications in reliability and storage. Springer, Heidelberg

    Book  MATH  Google Scholar 

  • Al Masry Z, Mercier S, Verdier G (2017) Approximate simulation techniques and distribution of an extended gamma process. Methodol Comput Appl Probab 19:213–235

    Article  MathSciNet  MATH  Google Scholar 

  • Atwood C (1986) The binomial failure rate common cause model. Technometrics 28(2):139–148

    Article  Google Scholar 

  • Barlow RE, Proschan F (1965) Mathematical theory of reliability, vol 17. Society for Industrial and Applied Mathematics (SIAM)

  • Barndorff-Nielsen O, Pedersen J, Sato K (2001) Multivariate subordination, self-decomposability and stability. Adv Appl Probab 33(1):160–187

    Article  MathSciNet  MATH  Google Scholar 

  • Cha JH, Mi J (2011) On a stochastic survival model for a system under randomly variable environment. Methodol Comput Appl Probab 13(3):549–561

    Article  MathSciNet  MATH  Google Scholar 

  • Çinlar E (1980) On a generalization of gamma processes. J Appl Probab 17(2):467–480

    Article  MathSciNet  MATH  Google Scholar 

  • Çınlar E (2011) Probability and stochastics. Graduate texts in mathematics, vol 261. Springer, New York

    MATH  Google Scholar 

  • Cont R, Tankov P (2004) Financial modelling with jump processes. Financial mathematics series. Chapman & Hall/CRC, Boca Raton

    MATH  Google Scholar 

  • Duffie D, Filipović D, Schachermayer W (2003) Affine processes and applications in finance. Ann Appl Probab 13(3):984–1053

    Article  MathSciNet  MATH  Google Scholar 

  • Dykstra R, Laud P (1981) A Bayesian nonparametric approach to reliability. Ann Stat 9(2):356–367

    Article  MathSciNet  MATH  Google Scholar 

  • Eberlein E, Kallsen J (2019) Mathematical finance. Springer Nature, Cham

    Book  MATH  Google Scholar 

  • Filipović D (2005) Time-inhomogeneous affine processes. Stoch Process Appl 115(4):639–659

    Article  MathSciNet  MATH  Google Scholar 

  • Guida M, Postiglione F, Pulcini G (2012) A time-discrete extended gamma process for time-dependent degradation phenomena. Reliab Eng Syst Saf 105:73–79

    Article  Google Scholar 

  • Jeanblanc M, Yor M, Chesney M (2009) Mathematical methods for financial markets. Springer, Berlin

    Book  MATH  Google Scholar 

  • Kallsen J, Tankov P (2006) Characterization of dependence of multidimensional Lévy processes using Lévy copulas. J Multivariate Anal 97(7):1551–1572

    Article  MathSciNet  MATH  Google Scholar 

  • Kebir Y (1991) On hazard rate processes. Nav Res Logist 38(6):865–876

    Article  MathSciNet  MATH  Google Scholar 

  • Kyprianou AE (2006) Introductory lectures on fluctuations of Lévy processes with applications. Universitext. Springer, Berlin

    MATH  Google Scholar 

  • Mallor F, Santos J (2003) Classification of shock models in system reliability. In: Seventh Zaragoza-Pau conference on applied mathematics and statistics (Spanish) (Jaca, 2001), vol 27 of Monogr. Semin. Mat. García Galdeano. Univ. Zaragoza, Zaragoza, pp 405–412 (2003)

  • Marshall AW, Olkin I (1967) A multivariate exponential distribution. J Am Stat Assoc 62(317):30–44

    Article  MathSciNet  MATH  Google Scholar 

  • Marshall AW, Shaked M (1979) Multivariate shock models for distributions with increasing hazard rate average. Ann Probab 7(2):343–358

    Article  MathSciNet  MATH  Google Scholar 

  • Mercier S, Pham HH (2016) A random shock model with mixed effect, including competing soft and sudden failures, and dependence. Methodol Comput Appl Probab 18(2):377–400

    Article  MathSciNet  MATH  Google Scholar 

  • Mercier S, Pham HH (2017) A bivariate failure time model with random shocks and mixed effects. J Multivar Anal 153:33–51

    Article  MathSciNet  MATH  Google Scholar 

  • Nelsen RB (2006) An introduction to copulas. Springer series in statistics, 2nd edn. Springer, New York

    Google Scholar 

  • Rausand M, Høyland A (2004) System reliability theory: models, statistical methods, and applications. Wiley series in probability and statistics, 2nd edn. Wiley, Hoboken

    MATH  Google Scholar 

  • Sato K (1999) Lévy processes and infinitely divisible distributions. Cambridge studies in advanced mathematics, vol 68. Cambridge University Press, Cambridge

    Google Scholar 

  • Singpurwalla ND, Youngren MA (1993) Multivariate distributions induced by dynamic environments. Scand J Stat 20(3):251–261

    MathSciNet  MATH  Google Scholar 

  • Vesely WE (1977) Nuclear systems reliability engineering and risk assessment, chapter estimating common cause failure probabilities in reliability and risk analysis: Marshall–Olkin specializations. United States: Society for Industrial and Applied Mathematics, pp 314–341

  • Wenocur ML (1989) A reliability model based on the gamma process and its analytic theory. Adv Appl Probab 21(4):899–918

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Both authors thank the Editor in Chief and the reviewers for their constructive comments on the model. This has lead us to better motivate the choice of the model and open the path for very interesting alternate models.

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Correspondence to Sophie Mercier.

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Mercier, S., Sangüesa, C. A general multivariate lifetime model with a multivariate additive process as conditional hazard rate increment process. Metrika 86, 91–129 (2023). https://doi.org/10.1007/s00184-022-00864-3

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  • DOI: https://doi.org/10.1007/s00184-022-00864-3

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