Objective Bayesian inference for the Capability index of the Weibull distribution and its generalization

The Weibull distribution plays an important role in reliability and quality control monitoring. This model has been widely used to describe the process capability index (PCI) when data do not follow a normal distribution. In this scenario, the current studies focus on estimating the parameters using classical inference. In this paper, we consider Bayesian methods to estimate the PCI denominated Cpk from an objective perspective using reference priors. The proposed inference is further extended to a generalized version of the Weibull distribution that provides a good fit for more complex data with non-monotone hazard behavior. The posterior distributions are constructed and Bayes estimators based on the median are proposed. In this case, Markov Chain Monte Carlo methods are used to achieve the estimates and from an extensive simulation study, we observe that good results are observed in terms of mean relative and squared errors. The proposed approach is also used to construct adequate credibility intervals with low computational cost and accurate coverage probabilities. A real data application is presented which confirms that our proposed approach outperforms the current methods.