Classical and Bayesian Inference for the New Lindley-X Time Model Under Random Beta-Binomial Progressive First-Failure Censoring: Theory, Computation, and Cancer Survival Applications

Incomplete lifetime observations arising from random subject withdrawals remain a major challenge in modern reliability and biomedical studies, where conventional progressive censoring schemes often fail to capture the stochastic nature of patient dropout and experimental interruption. Motivated by this limitation, this article develops a unified Bayesian inferential framework for the recently proposed one-parameter New XL (New-XL) lifetime distribution under a random beta-binomial progressive first-failure (B-Bin-PFF) censoring strategy. Both frequentist and Bayesian methodologies are developed for estimating the model parameter with the associated reliability and hazard rate functions. The derivation of maximum likelihood estimators includes two asymptotic confidence intervals, as well as Bayesian estimation accompanied by two credible intervals. Since the posterior distributions are analytically intractable, posterior inference is performed through a Metropolis–Hastings Markov chain Monte Carlo algorithm under independent gamma priors. An extensive Monte Carlo study investigates the finite-sample performance of the proposed procedures across a wide range of sample sizes, censoring proportions, group structures, and random withdrawal scenarios. The proposed methodology is finally illustrated using two clinically important cancer survival datasets involving gastric cancer and leukemia patients. The empirical analyses demonstrate that the proposed Bayesian framework provides accurate estimation, reliable uncertainty quantification, and superior goodness-of-fit compared with several competing lifetime models.