Bayesian Inference via Markov Iterative Methods for Generalized Progressive Hybrid Unit Bilal Censoring and Its Applications to Thermodynamics and Meteorology

The increasing availability of bounded lifetime observations in different disciplines has intensified the demand for flexible models capable of accommodating complex failure mechanisms. Motivated by this need, a comprehensive inferential framework is developed for the unit Bilal (UBilal) distribution using generalized progressive hybrid censoring, which guarantees a minimum number of observed failures while controlling experimental duration. Classical inference is established through maximum likelihood estimation, which is accompanied by asymptotic confidence intervals based on both normal and log-transformed approximations. Moreover, a Bayesian framework using a Metropolis–Hastings Markov chain Monte Carlo algorithm is presented. The proposed methodology further provides inference for important reliability characteristics, including the reliability and hazard rate functions, through both frequentist and Bayesian paradigms proposed. An extensive Monte Carlo investigation is conducted under diverse censoring schemes, sample sizes, and prior specifications to evaluate estimation accuracy, interval performance, and the influence of censoring severity. The simulation results show that Bayesian methods always provide better estimates and more reliable interval estimates, especially when prior information is used. Using two real datasets from thermodynamics and meteorology, the numerical results demonstrate that the UBilal model provides an excellent fit and yields reliable inference under bounded observations. Overall, the proposed methodology presents an efficient Bayesian inferential framework for bounded lifetime data collected through the generalized progressive hybrid censoring and expands the applicability of the UBilal model to reliability and related fields.