Objective Bayesian inference for the capability index of the Gamma distribution

The Gamma distribution has been applied in research in several areas of knowledge, due to its good flexibility and adaptability nature. Process capacity indices like Cpk are widely used when the measurements related to the data follow a normal distribution. This article aims to estimate the Cpk index for nonnormal data using the Gamma distribution. We discuss maximum likelihood estimation and a Bayesian analysis through the Gamma distribution using an objective prior, known as a matching prior that can return Bayesian estimates with good properties for the Cpk. A comparative study is made between classical and Bayesian estimation. The proposed Bayesian approach is considered with the Markov chain Monte Carlo method to generate samples of the posterior distribution. A simulation study is carried out to verify whether the posterior distribution presents good results when compared with the classical approach in terms of the mean relative errors and the mean square errors, which are the two commonly used metrics to evaluate the parameter estimators. Based on the real dataset, Bayesian estimates and credibility intervals for unknown parameters and the prior distribution are achieved to verify if the process is under control.