Objective Bayesian inference for the capability index of the Gamma distribution

Abstract

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.

Reliability assessment method for tank bottom plates based on hierarchical Bayesian corrosion growth model

“…The gamma process has the properties of non-negative, smooth growth, and independent incremental processes, which are widely used for the description of degradation processes such as wear and corrosion, and it has good mathematical properties for easy convolution processing. 24 So the gamma process is well suited for describing the monotonically growing process of corrosion defect growth at the bottom of the tank with a probability density function of :…”

Classical and objective Bayesian estimation and confidence intervals of an asymmetric loss based capability index Cpmk′$\mathcal {C}^{\prime }_{pmk}$

“…In this regard, readers may refer to the works of Arnold and Press 61 . In the absence of prior knowledge, non‐informative prior such as the Jeffreys prior may be considered for obtaining objective Bayesian analysis [see, de Almeida et al 43 …”

“…In this regard, readers may refer to the works of Arnold and Press 61 . In the absence of prior knowledge, non-informative prior such as the Jeffreys prior may be considered for obtaining objective Bayesian analysis [see, de Almeida et al 43 ]. For the problem at hand, we have considered objective Bayesian method with Jeffreys-rule prior for (𝜂, 𝜁 2 ), which is proportional to the square root of the determinant of the Fisher information matrix.…”

“…considered classical and Bayesian methods of estimation for the index $$\mathcal {C}_{py}$$ for the generalized Lindley distribution. de Almeida et al 43 . considered objective Bayesian analysis to estimate the PCI $$\mathcal {C}_{pk}$$ using the gamma distribution.…”

Statistical Inference on the Entropy Measures of Gamma Distribution under Progressive Censoring: EM and MCMC Algorithms

“…Basak and Balakrishnan [4] considered some classical procedures to estimate the three-parameter gamma distribution using progressive Type-II samples. Bayesian inference for the capability index of the gamma distribution was developed by Almeida et al [5]. Recently, Dey et al [6] derived maximum likelihood and Bayes estimates of the unknown parameters of the gamma lifetime model based on progressive Type-II censoring, and also studied the problem of interval estimation.…”