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Bayesian Inference of δ = P ( X < Y ) for Burr Type XII Distribution Based on Progressively First Failure-Censored Samples

Author

Listed:
  • Jessie Marie Byrnes

    (Department of Mathematical Sciences, University of South Dakota, Vermillion, SD 57069, USA)

  • Yu-Jau Lin

    (Department of Applied Mathematics, Chung Yuan Christian University, Chung Li District, Taoyuan City 32023, Taiwan)

  • Tzong-Ru Tsai

    (Department of Statistics, Tamkang University, Tamsui District, New Taipei City 25137, Taiwan)

  • Yuhlong Lio

    (Department of Mathematical Sciences, University of South Dakota, Vermillion, SD 57069, USA)

Abstract

Let X and Y follow two independent Burr type XII distributions and δ = P ( X < Y ) . If X is the stress that is applied to a certain component and Y is the strength to sustain the stress, then δ is called the stress–strength parameter. In this study, The Bayes estimator of δ is investigated based on a progressively first failure-censored sample. Because of computation complexity and no closed form for the estimator as well as posterior distributions, the Markov Chain Monte Carlo procedure using the Metropolis–Hastings algorithm via Gibbs sampling is built to collect a random sample of δ via the joint distribution of the progressively first failure-censored sample and random parameters and the empirical distribution of this collected sample is used to estimate the posterior distribution of δ . Then, the Bayes estimates of δ using the square error, absolute error, and linear exponential error loss functions are obtained and the credible interval of δ is constructed using the empirical distribution. An intensive simulation study is conducted to investigate the performance of these three types of Bayes estimates and the coverage probabilities and average lengths of the credible interval of δ . Moreover, the performance of the Bayes estimates is compared with the maximum likelihood estimates. The Internet of Things and a numerical example about the miles-to-failure of vehicle components for reliability evaluation are provided for application purposes.

Suggested Citation

  • Jessie Marie Byrnes & Yu-Jau Lin & Tzong-Ru Tsai & Yuhlong Lio, 2019. "Bayesian Inference of δ = P ( X < Y ) for Burr Type XII Distribution Based on Progressively First Failure-Censored Samples," Mathematics, MDPI, vol. 7(9), pages 1-24, September.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:9:p:794-:d:262851
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    References listed on IDEAS

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    4. Y. L. Lio & Tzong-Ru Tsai, 2012. "Estimation of δ= P ( X > Y ) for Burr XII distribution based on the progressively first failure-censored samples," Journal of Applied Statistics, Taylor & Francis Journals, vol. 39(2), pages 309-322, April.
    5. Chansoo Kim & Younshik Chung, 2006. "Bayesian estimation of P (Y>X) from Burr-type X model containing spurious observations," Statistical Papers, Springer, vol. 47(4), pages 643-651, October.
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    Cited by:

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