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Understanding the shape of the mixture failure rate (with engineering and demographic applications)

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  • Maxim Finkelstein

Abstract

Mixtures of distributions are usually effectively used for modelling heterogeneity. It is well known that mixtures of DFR distributions are always DFR. On the other hand, mixtures of IFR distributions can decrease, at least in some intervals of time. As IFR distributions often model lifetimes governed by ageing processes, the operation of mixing can dramatically change the pattern of ageing. Therefore, the study of the shape of the observed (mixture) failure rate in a heterogeneous setting is important in many applications. We study discrete and continuous mixtures, obtain conditions for the mixture failure rate to tend to the failure rate of the strongest populations and describe asymptotic behaviour as t→∞. Some demographic and engineering examples are considered. The corresponding inverse problem is discussed. Copyright © 2009 John Wiley & Sons, Ltd.

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  • Maxim Finkelstein, 2009. "Understanding the shape of the mixture failure rate (with engineering and demographic applications)," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 25(6), pages 643-663, November.
    • Handle: RePEc:wly:apsmbi:v:25:y:2009:i:6:p:643-663
    DOI: 10.1002/asmb.815
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    6. Eshetu T. Wondmagegnehu, 2004. "On the behavior and shape of mixture failure rates from a family of IFR Weibull distributions," Naval Research Logistics (NRL), John Wiley & Sons, vol. 51(4), pages 491-500, June.
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    1. Ji Hwan Cha & Maxim Finkelstein, 2012. "Burn-in and the performance quality measures in continuous heterogeneous populations," Journal of Risk and Reliability, , vol. 226(4), pages 417-425, August.
    2. Asadi, Majid & Ebrahimi, Nader & Soofi, Ehsan S., 2018. "Optimal hazard models based on partial information," European Journal of Operational Research, Elsevier, vol. 270(2), pages 723-733.
    3. Cha, Ji Hwan & Finkelstein, Maxim & Levitin, Gregory, 2021. "Optimal warranty policy with inspection for heterogeneous, stochastically degrading items," European Journal of Operational Research, Elsevier, vol. 289(3), pages 1142-1152.
    4. Finkelstein, Maxim, 2012. "On ordered subpopulations and population mortality at advanced ages," Theoretical Population Biology, Elsevier, vol. 81(4), pages 292-299.
    5. 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.
    6. M Shafiee & M Finkelstein & S Chukova, 2011. "Burn-in and imperfect preventive maintenance strategies for warranted products," Journal of Risk and Reliability, , vol. 225(2), pages 211-218, June.
    7. Maxim S. Finkelstein, 2011. "On ordered subpopulations and population mortality at advanced ages," MPIDR Working Papers WP-2011-022, Max Planck Institute for Demographic Research, Rostock, Germany.
    8. Lai, Chin-Diew & Izadi, Muhyiddin, 2012. "Generalized logistic frailty model," Statistics & Probability Letters, Elsevier, vol. 82(11), pages 1969-1977.
    9. Cha, Ji Hwan & Finkelstein, Maxim, 2011. "Burn-in and the performance quality measures in heterogeneous populations," European Journal of Operational Research, Elsevier, vol. 210(2), pages 273-280, April.

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