Reliability Inference Based on the Three-Parameter Burr Type XII Distribution with Type II Censoring

The three-parameter Burr type XII distribution (3pBXIID) is quite flexible and contains a wide range of distribution shapes for fitting lifetime data. However, it is difficult to obtain reliable estimates of the 3pBXIID quantiles from censored samples for evaluating the reliability of lifetime data. In this work, a Metropolis–Hastings Markov chain Monte Carlo (M-H MCMC) procedure is proposed to obtain reliable maximum likelihood estimates (MLEs) of the 3pBXIID quantiles from a type II censored sample. Moreover, the parametric bootstrap percentile procedure is used to obtain the confidence interval of the quantile of the 3pBXIID. The performance of the proposed M-H MCMC method is evaluated in view of Monte Carlo simulations. Two examples, regarding the survival lifetimes of breast cancer patients and the reliability inference on the lifetimes of oil-well pumps for sucker-rod oil pumping systems, are applied to illustrate the applications of the proposed M-H MCMC method and bootstrap procedure.