New Power Reliability Modeling via Randomized Progressive First-Failure Beta–Binomial Censoring: Theory, Optimization, and Engineering Applications to Fiber Strengths

In modern reliability engineering, modeling bounded lifetime data under realistic experimental conditions is still challenging, especially when censoring schemes and unit removals are random. This study proposes a new and unified reliability framework by combining the flexible powering new power (PNP) distribution with a grouping-based progressive first-failure mechanism using a beta-binomial random design. The proposed approach explicitly accounts for the randomness in group removals, providing a more realistic description of practical life-testing experiments. Classical estimation is carried out using maximum likelihood methods with the Newton-Raphson algorithm, along with confidence intervals constructed under both standard and log-transformed parameterizations. To increase flexibility in inference, a Bayesian approach is developed based on a joint gamma and shifted log-normal prior, which respects parameter constraints and incorporates prior uncertainty. Since the posterior distributions cannot be obtained in closed form, a Metropolis-Hastings Markov chain Monte Carlo algorithm is used to generate reliable posterior estimates and credible intervals. Additionally, beyond sensitivity analysis, multiple prior robustness diagnostics are incorporated to ensure reliable hyperparameter calibration and to safeguard against prior misspecification. The performance of the proposed estimators is carefully examined through extensive Monte Carlo simulations under different censoring schemes and parameter settings. The simulation results indicate that the proposed Bayesian procedures often provide more stable estimation and shorter interval estimates with competitive coverage probabilities compared with the corresponding classical methods, particularly under moderate-to-heavy censoring settings. To demonstrate its practical usefulness, the proposed model is applied to two real datasets on tensile strength of carbon and polyester fibers, where it provides a good fit and useful insights into material reliability and failure behavior. In the same applications, the practical relevance and superior performance of the proposed distribution are demonstrated, where it outperforms existing bounded versions of several well-known models, including the gamma, Weibull, and Birnbaum-Saunders distributions. Overall, this work contributes to reliability analysis by offering a flexible and computationally efficient framework that accounts for both random censoring and complex lifetime patterns, with potential applications in engineering, materials science, and applied reliability studies.