A Frailty-Augmented Weibull–Poisson Competing Risk Model for \(k\)-out-of-\(n\) System Reliability: Properties, Bayesian Estimation, and Simulation

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DOI: https://doi.org/10.28924/2291-8639-24-2026-358
Published: Oct 26, 2026
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Mohieddine Rahmouni, A Frailty-Augmented Weibull–Poisson Competing Risk Model for \(k\)-out-of-\(n\) System Reliability: Properties, Bayesian Estimation, and Simulation, Int. J. Anal. Appl., 24 (2026), 358.

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Mohieddine Rahmouni

Abstract

This paper introduces the Weibull–Poisson frailty (WPF) distribution, a four-parameter lifetime model in which a component fails at the earliest of the failure times of a zero-truncated Poisson number of latent causes, each cause having a Weibull failure time modulated by its own gamma frailty. The system lifetime is the \(k\)-th order statistic of \(n\) independent component lifetimes, so the model applies directly to \(k\)-out-of-\(n\) systems. The survival, density and hazard functions are available in closed form, and the model nests six distributions, including the Weibull–Poisson and gamma-frailty Weibull models. We characterize the possible hazard shapes (decreasing, increasing and unimodal), establish a stochastic ordering in the scale, competing-risk and frailty parameters, derive the condition under which the moments are finite, and show that ignoring the frailty while keeping the other parameters fixed always underestimates the mean time to failure (MTTF), most strongly for components with a decreasing hazard. Bayesian estimation is carried out with an adaptive Metropolis-within-Gibbs sampler. A Monte Carlo study shows that the Bayesian estimator recovers the shape parameter and the MTTF accurately and is far more stable than maximum likelihood, whereas the scale and competing-risk parameters are only weakly identified. Two applications illustrate the model: the frailty component is not supported by Aarset’s device-lifetime data, but it significantly improves the fit to Badar and Priest’s carbon fiber strengths.

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References

  1. W. Kuo, M.J. Zuo, Optimal Reliability Modeling: Principles and Applications, John Wiley & Sons, 2003.
  2. R.E. Barlow, F. Proschan, Statistical Theory of Reliability and Life Testing, Holt, Rinehart and Winston, 1975.
  3. K. Adamidis, S. Loukas, A Lifetime Distribution with Decreasing Failure Rate, Stat. Probab. Lett. 39 (1998), 35–42. https://doi.org/10.1016/s0167-7152(98)00012-1.
  4. C. Kuş, A New Lifetime Distribution, Comput. Stat. Data Anal. 51 (2007), 4497–4509. https://doi.org/10.1016/j.csda.2006.07.017.
  5. W. Barreto-Souza, A.L. de Morais, G.M. Cordeiro, The Weibull-Geometric Distribution, J. Stat. Comput. Simul. 81 (2011), 645–657. https://doi.org/10.1080/00949650903436554.
  6. E.M.P. Bereta, F. Louzada, M.A.P. Franco, The Poisson-Weibull Distribution, Adv. Appl. Stat. 22 (2011), 107–118.
  7. W. Lu, D. Shi, A New Compounding Life Distribution: The Weibull–Poisson Distribution, J. Appl. Stat. 39 (2012), 21–38. https://doi.org/10.1080/02664763.2011.575126.
  8. A.L. Morais, W. Barreto-Souza, A Compound Class of Weibull and Power Series Distributions, Comput. Stat. Data Anal. 55 (2011), 1410–1425. https://doi.org/10.1016/j.csda.2010.09.030.
  9. M.H. Tahir, G.M. Cordeiro, Compounding of Distributions: A Survey and New Generalized Classes, J. Stat. Distrib. Appl. 3 (2016), 13. https://doi.org/10.1186/s40488-016-0052-1.
  10. M. Rahmouni, D. Ziedan, The Weibull-Generalized Shifted Geometric Distribution: Properties, Estimation, and Applications, AIMS Math. 10 (2025), 9773–9804. https://doi.org/10.3934/math.2025448.
  11. M. Rahmouni, A New Compound Family of Weibull Order Statistics and Left k-Truncated Power Series Distributions, Int. J. Anal. Appl. 23 (2025), 106. https://doi.org/10.28924/2291-8639-23-2025-106.
  12. M. Rahmouni, The Doubly Generalized Exponential-Geometric Frailty Distribution, AIMS Math. 10 (2025), 30384–30428. https://doi.org/10.3934/math.20251334.
  13. M. Rahmouni, From Exponential-Geometric to Lomax: A Unified Survival Model via Gamma Frailty, AIMS Math. 10 (2025), 29927–29954. https://doi.org/10.3934/math.20251315.
  14. M. Rahmouni, A Logarithmic–Gamma Frailty Model for First-Failure Times with Heavy-Tailed Risk, AIMS Math. 11 (2026), 8945–8965. https://doi.org/10.3934/math.2026369.
  15. M. Rahmouni, The Doubly Generalized Weibull Power Series Frailty Distribution, Int. J. Anal. Appl. 23 (2025), 255. https://doi.org/10.28924/2291-8639-23-2025-255.
  16. J.W. Vaupel, K.G. Manton, E. Stallard, The Impact of Heterogeneity in Individual Frailty on the Dynamics of Mortality, Demography 16 (1979), 439–454. https://doi.org/10.2307/2061224.
  17. P. Hougaard, Analysis of Multivariate Survival Data, Springer New York, 2000. https://doi.org/10.1007/978-1-4612-1304-8.
  18. G.O. Roberts, J.S. Rosenthal, Examples of Adaptive MCMC, J. Comput. Graph. Stat. 18 (2009), 349–367. https://doi.org/10.1198/jcgs.2009.06134.
  19. A. Gelman, D.B. Rubin, Inference from Iterative Simulation Using Multiple Sequences, Stat. Sci. 7 (1992), 457–472. https://doi.org/10.1214/ss/1177011136.
  20. A. Gelman, J.B. Carlin, H.S. Stern, D.B. Dunson, A. Vehtari, et al., Bayesian Data Analysis, Chapman and Hall/CRC, 2013. https://doi.org/10.1201/b16018.
  21. M.V. Aarset, How to Identify a Bathtub Hazard Rate, IEEE Trans. Reliab. R-36 (1987), 106–108. https://doi.org/10.1109/tr.1987.5222310.
  22. G.S. Mudholkar, D.K. Srivastava, Exponentiated Weibull Family for Analyzing Bathtub Failure-Rate Data, IEEE Trans. Reliab. 42 (1993), 299–302. https://doi.org/10.1109/24.229504.
  23. M. Xie, C.D. Lai, Reliability Analysis Using an Additive Weibull Model with Bathtub-Shaped Failure Rate Function, Reliab. Eng. Syst. Saf. 52 (1996), 87–93. https://doi.org/10.1016/0951-8320(95)00149-2.
  24. M.G. Badar, A.M. Priest, Statistical Aspects of Fiber and Bundle Strength in Hybrid Composites, in: T. Hayashi, K. Kawata, S. Umekawa, (Eds) Progress in Science and Engineering of Composites, ICCM-IV, pp. 1129–1136, 1982.