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

Jessie Marie Byrnes, Yu-Jau Lin, Tzong-Ru Tsai and Yuhlong Lio
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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

Mathematics, 2019, vol. 7, issue 9, 1-24

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.

Keywords: gibbs sampling; Markov Chain Monte Carlo; maximum likelihood estimation; Metropolis–Hastings algorithm; progressive first failure-censoring scheme (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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