Hybrid Firefly–Bayesian Neural Network Estimation for Reliable Modeling of the Inverse Gompertz–Gumbel Family and NHPP Systems: Properties and Comparative Evaluation

The idea commonly accepted is that probability distributions play a major role in reliability modeling, whereas classical models are often not flexible enough to accommodate the complex behavior of life data. This paper presents a more flexible modeling approach to reliability analysis through the introduction of the Inverse Gompertz Gumbel (IGoGum) distribution and Non-Homogeneous Poisson Process (IGoGum NHPP). The IGoGum-NHPP model, which is derived from the reciprocal transformation of the Gompertz-Gumbel distribution, accommodates skewness in positive and negative directions and allows for increasing and decreasing forms of hazard. The essential statistical properties of the model, including the survival and hazard functions, moments and order statistics, and the entropy of Bayesian risk, will be formulated along with parameter estimation through maximum likelihood, Bayesian, and hybrid Bayesian Neural Networks being optimized through Firefly Algorithms (BNN–FFA). Monte Carlo simulation and other model selection criteria (AIC, BIC, CAIC, HQIC, KS, MSE, RMSE) demonstrate that the proposed estimators perform very efficiently and to great effect. Applications to bladder-cancer remission time’s data and a data set regarding failures of diesel-engine turbochargers revealed that the IGoGum -NHPP model outperforms the others in terms of the fit and predictive capacity (invert Gompertz -Frankfurt -Vredenburg, Gompertz -Burr XII, and Gompertz -Lomax). Thus, the IGoGum family becomes a multifunctional and efficient tool to characterize lifetime modeling, reliability evaluation, and risk identification of complex systems.