Statistical Evaluation of Beta-Binomial Probability Law for Removal in Progressive First-Failure Censoring and Its Applications to Three Cancer Cases

Progressive first-failure censoring is a flexible and cost-efficient strategy that captures real-world testing scenarios where only the first failure is observed at each stage while randomly removing remaining units, making it ideal for biomedical and reliability studies. By applying the α -power transformation to the exponential baseline, the proposed model introduces an additional flexibility parameter that enriches the family of lifetime distributions, enabling it to better capture varying failure rates and diverse hazard rate behaviors commonly observed in biomedical data, thus extending the classical exponential model. This study develops a novel computational framework for analyzing an α -powered exponential model under beta-binomial random removals within the proposed censoring test. To address the inherent complexity of the likelihood function arising from simultaneous random removals and progressive censoring, we derive closed-form expressions for the likelihood, survival, and hazard functions and propose efficient estimation strategies based on both maximum likelihood and Bayesian inference. For the Bayesian approach, gamma and beta priors are adopted, and a tailored Metropolis–Hastings algorithm is implemented to approximate posterior distributions under symmetric and asymmetric loss functions. To evaluate the empirical performance of the proposed estimators, extensive Monte Carlo simulations are conducted, examining bias, mean squared error, and credible interval coverage under varying censoring levels and removal probabilities. Furthermore, the practical utility of the model is illustrated through three oncological datasets, including multiple myeloma, lung cancer, and breast cancer patients, demonstrating superior goodness of fit and predictive reliability compared to traditional models. The results show that the proposed lifespan model, under the beta-binomial probability law and within the examined censoring mechanism, offers a flexible and computationally tractable framework for reliability and biomedical survival analysis, providing new insights into censored data structures with random withdrawals.