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 $C_{pk}$ are widely used when the measurements related to the data follow a normal distribution. This article aims to estimate the $C_{pk}$ 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 $C_{pk}$. 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.

中文翻译：

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Gamma 分布能力指数的客观贝叶斯推断

Gamma 分布由于其良好的灵活性和适应性，已被应用于多个知识领域的研究。处理能力指数，如$C_{磷克}$当与数据相关的测量值服从正态分布时被广泛使用。本文旨在估计$C_{磷克}$使用 Gamma 分布的非正态数据索引。我们使用客观先验（称为匹配先验）通过 Gamma 分布讨论最大似然估计和贝叶斯分析，它可以返回具有良好属性的贝叶斯估计$C_{磷克}$. 对经典估计和贝叶斯估计进行了比较研究。建议的贝叶斯方法与马尔可夫链蒙特卡罗方法一起考虑以生成后验分布的样本。进行模拟研究以验证后验分布与经典方法相比是否在平均相对误差和均方误差方面表现出良好的结果，这是评估参数估计量的两个常用指标。基于真实数据集，实现未知参数的贝叶斯估计和可信区间以及先验分布，以验证过程是否受控。