Inference for Two-Parameter Birnbaum–Saunders Distribution Under Joint Progressively Type-II Censored Data

We study inference and prediction for two populations whose lifetimes follow two-parameter Birnbaum–Saunders distributions under a joint progressive Type-II censoring scheme. We derive the observed-data likelihood and obtain maximum likelihood estimates via an EM algorithm that treats progressively removed lifetimes as missing data. Bayesian inference is developed using importance sampling and a hybrid Gibbs–Metropolis–Hastings sampler, leading to Bayes estimators, credible intervals, and posterior predictive summaries. We further construct prediction intervals for the unobserved lifetimes removed at multiple censoring stages. Monte Carlo experiments under several censoring patterns and parameter configurations compare the frequentist and Bayesian procedures. A tuberculosis survival dataset illustrates model adequacy, parameter estimation, and prediction of removed units under joint progressive censoring.