A Bayesian inference with Hamiltonian Monte Carlo (HMC) framework for a three-parameter model with reliability applications

In this work, a complete Bayesian paradigm for the proposed three-parameter Weibull-based model is presented, and the Hamiltonian Monte Carlo (HMC) algorithm was used to enhance precision and expedite inference. Simulation studies were used to evaluate the appropriateness of the proposed Bayes estimators. In addition, maximum likelihood estimators (MLEs) are also presented. We demonstrate that the MLEs for each parameter exist under certain conditions, with some being uniquely identifiable. Moreover, comprehensive reliability characteristics of the proposed model were derived and studied, such as the reliability function, failure rate function, mean residual life, and $r$th moments. We also investigated the identifiability of the proposed model’s parameters. Finally, two real datasets involving the failure times of some components were used to evaluate the performance of the proposed estimation methods and the model. The proposed model outperformed many existing models, ranking first in both dataset evaluations by consistently achieving more of the lowest values in the Akaike information criterion (AIC), Bayesian information criterion, corrected AIC, Kolmogorov–Smirnov test, Anderson–Darling test, and Cramér–von Mises test.