Comprehensive Assessments of the Bilal Extended Model with Applications in Mechanical Engineering and Health Insurance

A recent generalized Bilal (G-Bilal) model demonstrates remarkable flexibility in capturing a wide spectrum of failure behaviors, including monotonic and non-monotonic (upside-down bathtub-shaped) hazard patterns, outperforming several existing models such as the Weibull, gamma, and exponential families. This paper develops several inferential frameworks for different G-Bilal parameters of life using samples gathered by improved Type-II adaptive progressive censoring. This enhanced design ensures optimal control of test duration while maintaining high inferential precision. Expressions for the model parameters, reliability, and hazard rate functions are derived, followed by the development of maximum likelihood (ML) and maximum product of spacing (MPS) estimators with their asymptotic confidence intervals using the observed Fisher information with the delta approach. Furthermore, Bayesian estimators and two associated credible intervals are proposed under independent gamma priors and computed through Markov iterations, with both ML and MPS posteriors considered. Extensive Monte Carlo experiments confirm the consistency, robustness, and precision of the proposed estimators, with Bayesian spacing-based methods exhibiting superior accuracy and coverage. The model’s practical potential is further verified through two real applications: one involving mechanical system lifetimes and another analyzing health insurance premium data, representing physical and actuarial domains, respectively. Using the introduced censoring, the proposed G-Bilal model outperforms all competing models in terms of goodness-of-fit and reliability estimates in both cases. The results underscore the G-Bilal model’s adaptability, computational stability, and empirical superiority, establishing it as a powerful tool for modern reliability and actuarial risk assessments.