Abstract
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.
Highlights
Because the likelihood function based on accelerated life test (ALT) samples from the doubly truncated three-parameter Burr type XII (BurrXII) distribution is very complicated, the explicit forms of maximum likelihood estimators for the model parameters cannot be derived
Let the probability density function (PDF) and cumulative distribution function (CDF) of a reliable unit lifetime, X, be f ( x |θ ) and F ( x |θ ), respectively, where θ is the vector of distribution parameters
The doubly truncated three-parameter BurrXII distribution is used to model the lifetimes of reliable units and the lifetimes of units are collected under the constant-stress ALT method to save the test time and sample resources
Summary
Under the progressive type II censoring, Abdel-Hamid [21] studied the maximum likelihood estimation method and provided a Fisher information matrix to implement reliability inferences for BurrXII distributions under the constant-partially accelerated life testing. Zhao et al [23] proposed a simple constant-stress ALT to collect type I progressively hybrid censored samples from BurrXII distribution and obtained the maximum likelihood estimators of the model parameters via using numerical methods. They obtained approximate confidence intervals of the distribution parameters through using the normal approximation and bootstrap methods. Afify et al [36] proposed maximum likelihood estimation and Bayesian estimation methods to estimate the parameters of the generalized odd log-logistic exponential distribution
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