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Bayesian estimation versus maximum likelihood estimation in the Weibull-power law process

Author

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  • Alicja Jokiel-Rokita

    (Wroclaw University of Science and Technology to Wrocław University of Science and Technology)

  • Ryszard Magiera

    (Wroclaw University of Science and Technology to Wrocław University of Science and Technology)

Abstract

The Bayesian approach is applied to estimation of the Weibull-power law process (WPLP) parameters as an alternative to the maximum likelihood (ML) method in the case when the number of events is small. For the process model considered we propose to apply the independent Jeffreys prior distribution and we argue that this is a useful choice. Comparisons were also made between the accuracy of the estimators obtained and those obtained by using other priors—informative and weakly informative. The investigations show that the Bayesian approach in many cases of a fairly broad collection of WPLP models can lead to the Bayes estimators that are more accurate than the corresponding ML ones, when the number of events is small. The problem of fitting the WPLP models, based on ML and Bayes estimators, to some real data is also considered. It is shown that the TTT-concept, used in the reliability theory, is not fully useful for the WPLP models, and it may be so for some other trend-renewal processes. In order to assess the accuracy of the fitting to the real data considered, two other graphical methods are introduced.

Suggested Citation

  • Alicja Jokiel-Rokita & Ryszard Magiera, 2023. "Bayesian estimation versus maximum likelihood estimation in the Weibull-power law process," Computational Statistics, Springer, vol. 38(2), pages 675-710, June.
    • Handle: RePEc:spr:compst:v:38:y:2023:i:2:d:10.1007_s00180-022-01241-4
    DOI: 10.1007/s00180-022-01241-4
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    References listed on IDEAS

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    1. Chris Bambey Guure & Noor Akma Ibrahim & Al Omari Mohammed Ahmed, 2012. "Bayesian Estimation of Two-Parameter Weibull Distribution Using Extension of Jeffreys' Prior Information with Three Loss Functions," Mathematical Problems in Engineering, Hindawi, vol. 2012, pages 1-13, August.
    2. Shaul Bar-Lev & Idit Lavi & Benjamin Reiser, 1992. "Bayesian inference for the power law process," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 44(4), pages 623-639, December.
    3. Bo Bergman & Bengt Klefsjö, 1984. "The Total Time on Test Concept and Its Use in Reliability Theory," Operations Research, INFORMS, vol. 32(3), pages 596-606, June.
    4. M. Aminzadeh, 2013. "Bayesian estimation of the expected time of first arrival past a truncated time T: the case of NHPP with power law intensity," Computational Statistics, Springer, vol. 28(6), pages 2465-2477, December.
    5. Nibedita Bandyopadhyay & Ananda Sen, 2005. "Non-standard asymptotics in an inhomogeneous gamma process," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 57(4), pages 703-732, December.
    6. Haigang Zhou & Steven Rigdon, 2008. "Duration dependence in US business cycles: An analysis using the modulated power law process," Journal of Economics and Finance, Springer;Academy of Economics and Finance, vol. 32(1), pages 25-34, January.
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