Adaptive estimation: Fuzzy data-driven gamma distribution via Bayesian and maximum likelihood approaches

Integrating fuzzy concepts into statistical estimation offers considerable advantages by enhancing both the accuracy and reliability of parameter estimations, irrespective of the sample size and technique used. This study specifically examined the improvement of parameter estimation accuracy when dealing with fuzzy data, with a focus on the gamma distribution. We explored and evaluated a variety of estimation techniques for determining the scale parameter $\eta$ and shape parameter $\rho$ of the gamma distribution, employing both maximum likelihood (ML) and Bayesian methods. In the case of ML estimates, the expectation-maximization (EM) algorithm and the Newton-Raphson (NR) method were applied, with confidence intervals constructed using the Fisher information matrix. Additionally, the highest posterior density (HPD) intervals were derived through Gibbs sampling. For Bayesian estimates, the Tierney and Kadane (TK) approximation and Gibbs sampling were used to enhance the estimation process. A thorough performance comparison was undertaken using a simulated fuzzy dataset of the lifetimes of rechargeable batteries to assess the effectiveness of these methods. The methods were evaluated by comparing the estimated parameters to their true values using mean squared error (MSE) as a metric. Our findings demonstrate that the Bayesian approach, particularly when combined with the TK method, consistently produces more accurate and reliable parameter estimates compared to traditional methods. These results underscore the potential of Bayesian techniques in addressing fuzzy data and enhancing precision in statistical analyses.