Fuzzy Data Modeling and Parameter Estimation in Two Gamma Populations

This study addresses the challenge of estimating parameters for two Gamma populations that share a common scale parameter but differ in their shape parameters, within the context of fuzzy data. To manage these complexities, both Maximum Likelihood and Bayesian estimation techniques are employed. Due to the absence of closed-form solutions for the Maximum Likelihood estimators, the Expectation- Maximization algorithm is utilized, and asymptotic confidence intervals are constructed based on the observed information matrix. For Bayesian estimation, a conjugate prior is used to derive Bayes estimators, which are approximated using Lindley’s method in light of the analytical intractability. Additionally, Gibbs sampling is implemented to estimate posterior densities and construct Highest Posterior Density intervals. Approximate Bayesian Computation is also employed as a likelihood-free approach to Bayesian inference, particularly useful under fuzzy data conditions where the likelihood is difficult to specify explicitly. A comprehensive comparison of Maximum Likelihood Estimation, Lindley’s approximation, Approximate Bayesian Computation, and Gibbs sampling is conducted to evaluate their performance. The effectiveness of the proposed methods is further illustrated using real data from a light-emitting diode manufacturing process.