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Some notes on unobserved parameters (frailties) in reliability modeling

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  • Cha, Ji Hwan
  • Finkelstein, Maxim

Abstract

Unobserved random quantities (frailties) often appear in various reliability problems especially when dealing with the failure rates of items from heterogeneous populations. As the failure rate is a conditional characteristic, the distributions of these random quantities, similar to Bayesian approaches, are updated in accordance with the corresponding survival information. At some instances, apart from a statistical meaning, frailties can have also useful interpretations describing the underlying lifetime model. We discuss and clarify these issues in reliability context and present and analyze several meaningful examples. We consider the proportional hazards model with a random factor; the stress–strength model, where the unobserved strength of a system can be viewed as frailty; a parallel system with a random number of components and, finally, the first passage time problem for the Wiener process with random parameters.

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  • Cha, Ji Hwan & Finkelstein, Maxim, 2014. "Some notes on unobserved parameters (frailties) in reliability modeling," Reliability Engineering and System Safety, Elsevier, vol. 123(C), pages 99-103.
    • Handle: RePEc:eee:reensy:v:123:y:2014:i:c:p:99-103
    DOI: 10.1016/j.ress.2013.10.008
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    References listed on IDEAS

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    1. Maxim Finkelstein & Ji Hwan Cha, 2013. "Burn-in for Heterogeneous Populations," Springer Series in Reliability Engineering, in: Stochastic Modeling for Reliability, edition 127, chapter 0, pages 261-312, Springer.
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    4. Cha, Ji Hwan & Finkelstein, Maxim, 2013. "The failure rate dynamics in heterogeneous populations," Reliability Engineering and System Safety, Elsevier, vol. 112(C), pages 120-128.
    5. Finkelstein, Maxim, 2012. "On ordered subpopulations and population mortality at advanced ages," Theoretical Population Biology, Elsevier, vol. 81(4), pages 292-299.
    6. Finkelstein, Maxim, 2007. "Shocks in homogeneous and heterogeneous populations," Reliability Engineering and System Safety, Elsevier, vol. 92(5), pages 569-574.
    7. Missov, Trifon I. & Finkelstein, Maxim, 2011. "Admissible mixing distributions for a general class of mixture survival models with known asymptotics," Theoretical Population Biology, Elsevier, vol. 80(1), pages 64-70.
    8. Maxim Finkelstein & Ji Hwan Cha, 2013. "Shocks as Burn-in," Springer Series in Reliability Engineering, in: Stochastic Modeling for Reliability, edition 127, chapter 0, pages 313-361, Springer.
    9. James Vaupel & Kenneth Manton & Eric Stallard, 1979. "The impact of heterogeneity in individual frailty on the dynamics of mortality," Demography, Springer;Population Association of America (PAA), vol. 16(3), pages 439-454, August.
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    Cited by:

    1. Almeida, Marco Pollo & Paixão, Rafael S. & Ramos, Pedro L. & Tomazella, Vera & Louzada, Francisco & Ehlers, Ricardo S., 2020. "Bayesian non-parametric frailty model for dependent competing risks in a repairable systems framework," Reliability Engineering and System Safety, Elsevier, vol. 204(C).
    2. Gregory Levitin & Maxim Finkelstein, 2017. "A new stress–strength model for systems subject to stochastic shocks," Journal of Risk and Reliability, , vol. 231(2), pages 172-179, April.
    3. Zeynab Allahkarami & Ahmad Reza Sayadi & Behzad Ghodrati, 2021. "Identifying the mixed effects of unobserved and observed risk factors on the reliability of mining hauling system," International Journal of System Assurance Engineering and Management, Springer;The Society for Reliability, Engineering Quality and Operations Management (SREQOM),India, and Division of Operation and Maintenance, Lulea University of Technology, Sweden, vol. 12(2), pages 281-289, April.
    4. Hooti, Fatemeh & Ahmadi, Jafar & Longobardi, Maria, 2020. "Optimal extended warranty length with limited number of repairs in the warranty period," Reliability Engineering and System Safety, Elsevier, vol. 203(C).
    5. Cha, Ji Hwan & Finkelstein, Maxim & Levitin, Gregory, 2018. "Optimal mission abort policy for partially repairable heterogeneous systems," European Journal of Operational Research, Elsevier, vol. 271(3), pages 818-825.
    6. Hao Zeng & Xuxue Sun & Kuo Wang & Yuxin Wen & Wujun Si & Mingyang Li, 2024. "A Bayesian Approach for Lifetime Modeling and Prediction with Multi-Type Group-Shared Missing Covariates," Mathematics, MDPI, vol. 12(5), pages 1-23, February.
    7. Rezgar Zaki & Abbas Barabadi & Javad Barabady & Ali Nouri Qarahasanlou, 2022. "Observed and unobserved heterogeneity in failure data analysis," Journal of Risk and Reliability, , vol. 236(1), pages 194-207, February.
    8. M. Kayid & S. Izadkhah & Ming J. Zuo, 2017. "Some results on the relative ordering of two frailty models," Statistical Papers, Springer, vol. 58(2), pages 287-301, June.
    9. Rezgar Zaki & Abbas Barabadi & Ali Nouri Qarahasanlou & A. H. S. Garmabaki, 2019. "A mixture frailty model for maintainability analysis of mechanical components: a case study," International Journal of System Assurance Engineering and Management, Springer;The Society for Reliability, Engineering Quality and Operations Management (SREQOM),India, and Division of Operation and Maintenance, Lulea University of Technology, Sweden, vol. 10(6), pages 1646-1653, December.
    10. Slimacek, Vaclav & Lindqvist, Bo Henry, 2016. "Nonhomogeneous Poisson process with nonparametric frailty," Reliability Engineering and System Safety, Elsevier, vol. 149(C), pages 14-23.
    11. Ramesh Gupta, 2016. "Properties of additive frailty model in survival analysis," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 79(1), pages 1-17, January.
    12. Asfaw, Zeytu Gashaw & Lindqvist, Bo Henry, 2015. "Unobserved heterogeneity in the power law nonhomogeneous Poisson process," Reliability Engineering and System Safety, Elsevier, vol. 134(C), pages 59-65.
    13. Awat Ghomghaleh & Reza Khaloukakaie & Mohammad Ataei & Abbas Barabadi & Ali Nouri Qarahasanlou & Omeid Rahmani & Amin Beiranvand Pour, 2020. "Prediction of remaining useful life (RUL) of Komatsu excavator under reliability analysis in the Weibull-frailty model," PLOS ONE, Public Library of Science, vol. 15(7), pages 1-16, July.
    14. J. Jarrahiferiz & M. Kayid & S. Izadkhah, 2019. "Stochastic properties of a weighted frailty model," Statistical Papers, Springer, vol. 60(1), pages 53-72, February.

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