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Accelerated Life Test Method for the Doubly Truncated Burr Type XII Distribution

1
School of Mathematics and Statistics, Northeast Petroleum University, Daqing 163318, Heilongjiang, China
2
Department of Mathematical Sciences, University of South Dakota, Vermillion, SD 57069, USA
3
Department of Statistics, Tamkang University, Tamsui District, New Taipei City 25137, Taiwan
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(2), 162; https://doi.org/10.3390/math8020162
Received: 30 December 2019 / Revised: 20 January 2020 / Accepted: 20 January 2020 / Published: 23 January 2020
(This article belongs to the Special Issue Statistical Simulation and Computation)
The Burr type XII (BurrXII) distribution is very flexible for modeling and has earned much attention in the past few decades. In this study, the maximum likelihood estimation method and two Bayesian estimation procedures are investigated based on constant-stress accelerated life test (ALT) samples, which are obtained from the doubly truncated three-parameter BurrXII distribution. Because computational difficulty occurs for maximum likelihood estimation method, two Bayesian procedures are suggested to estimate model parameters and lifetime quantiles under the normal use condition. A Markov Chain Monte Carlo approach using the Metropolis–Hastings algorithm via Gibbs sampling is built to obtain Bayes estimators of the model parameters and to construct credible intervals. The proposed Bayesian estimation procedures are simple for practical use, and the obtained Bayes estimates are reliable for evaluating the reliability of lifetime products based on ALT samples. Monte Carlo simulations were conducted to evaluate the performance of these two Bayesian estimation procedures. Simulation results show that the second Bayesian estimation procedure outperforms the first Bayesian estimation procedure in terms of bias and mean squared error when users do not have sufficient knowledge to set up hyperparameters in the prior distributions. Finally, a numerical example about oil-well pumps is used for illustration. View Full-Text
Keywords: accelerated life test; Burr type XII distribution; Markov chain Monte Carlo; maximum likelihood estimation; Newton–Raphson method accelerated life test; Burr type XII distribution; Markov chain Monte Carlo; maximum likelihood estimation; Newton–Raphson method
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Xin, H.; Liu, Z.; Lio, Y.; Tsai, T.-R. Accelerated Life Test Method for the Doubly Truncated Burr Type XII Distribution. Mathematics 2020, 8, 162.

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