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The discrete additive Weibull distribution: A bathtub-shaped hazard for discontinuous failure data

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  • Bebbington, Mark
  • Lai, Chin-Diew
  • Wellington, Morgan
  • Zitikis, RiÄ ardas

Abstract

Although failure data are usually treated as being continuous, they may have been collected in a discrete manner, or in fact be discrete in nature. Reliability models with bathtub-shaped hazard rate are fundamental to the concepts of burn-in and maintenance, but how well do they incorporate discrete data? We explore discrete versions of the additive Weibull distribution, which has the twin virtues of mathematical tractability and the ability to produce bathtub-shaped hazard rate functions. We derive conditions on the parameters for the hazard rate function to be increasing, decreasing, or bathtub shaped. While discrete versions may have the same shaped hazard rate for the same parameter values, we find that when fitted to data the fitted hazard rate shapes can vary between versions. Our results are illustrated using several real-life data sets, and the implications of using continuous models for discrete data discussed.

Suggested Citation

  • Bebbington, Mark & Lai, Chin-Diew & Wellington, Morgan & Zitikis, RiÄ ardas, 2012. "The discrete additive Weibull distribution: A bathtub-shaped hazard for discontinuous failure data," Reliability Engineering and System Safety, Elsevier, vol. 106(C), pages 37-44.
    • Handle: RePEc:eee:reensy:v:106:y:2012:i:c:p:37-44
    DOI: 10.1016/j.ress.2012.06.009
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    References listed on IDEAS

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    1. Bebbington, Mark & Lai, Chin-Diew & Zitikis, RiÄ ardas, 2007. "A flexible Weibull extension," Reliability Engineering and System Safety, Elsevier, vol. 92(6), pages 719-726.
    2. Carrasco, Jalmar M.F. & Ortega, Edwin M.M. & Cordeiro, Gauss M., 2008. "A generalized modified Weibull distribution for lifetime modeling," Computational Statistics & Data Analysis, Elsevier, vol. 53(2), pages 450-462, December.
    3. Jiang, R., 2010. "Discrete competing risk model with application to modeling bus-motor failure data," Reliability Engineering and System Safety, Elsevier, vol. 95(9), pages 981-988.
    4. Mark Bebbington & Chin-Diew Lai & Ričardas Zitikis, 2007. "Optimum Burn-in Time for a Bathtub Shaped Failure Distribution," Methodology and Computing in Applied Probability, Springer, vol. 9(1), pages 1-20, March.
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    5. Robab Aghazadeh Chakherloo & Mohammad Pourgol-Mohammad & Kamran Sepanloo, 2017. "Change points estimations of bathtub-shaped hazard functions," 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. 8(3), pages 553-559, September.
    6. Hanan Haj Ahmad & Dina A. Ramadan & Ehab M. Almetwally, 2024. "Evaluating the Discrete Generalized Rayleigh Distribution: Statistical Inferences and Applications to Real Data Analysis," Mathematics, MDPI, vol. 12(2), pages 1-23, January.
    7. M. S. Eliwa & Ziyad Ali Alhussain & M. El-Morshedy, 2020. "Discrete Gompertz-G Family of Distributions for Over- and Under-Dispersed Data with Properties, Estimation, and Applications," Mathematics, MDPI, vol. 8(3), pages 1-26, March.
    8. Mohamed S. Eliwa & Mahmoud El-Morshedy & Haitham M. Yousof, 2022. "A Discrete Exponential Generalized-G Family of Distributions: Properties with Bayesian and Non-Bayesian Estimators to Model Medical, Engineering and Agriculture Data," Mathematics, MDPI, vol. 10(18), pages 1-29, September.
    9. Mohamed Aboraya & Haitham M. Yousof & G.G. Hamedani & Mohamed Ibrahim, 2020. "A New Family of Discrete Distributions with Mathematical Properties, Characterizations, Bayesian and Non-Bayesian Estimation Methods," Mathematics, MDPI, vol. 8(10), pages 1-25, September.
    10. Mohamed Ibrahim & M. Masoom Ali & Haitham M. Yousof, 2023. "The Discrete Analogue of the Weibull G Family: Properties, Different Applications, Bayesian and Non-Bayesian Estimation Methods," Annals of Data Science, Springer, vol. 10(4), pages 1069-1106, August.
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