Entropy-Regularized Likelihood Inference for Lifetime Distributions Under Progressive Type-II Censoring

This study proposes an entropy-regularized likelihood inference framework for lifetime distributions under progressive Type-II censoring. By incorporating Shannon entropy directly into the classical likelihood function, the proposed approach aims to alleviate the information loss caused by censoring and improve the finite-sample stability of parameter estimation. Entropy-regularized maximum likelihood estimators (ERMLEs) are developed for the Exponential, Weibull, Gamma, and Lognormal lifetime distributions. Distribution-specific regularization parameters are selected by minimizing the average mean squared error across a comprehensive Monte Carlo simulation study covering different sample sizes, censoring rates, and progressive censoring schemes. Estimation performance is evaluated using bias, mean squared error, and the relative reduction in MSE achieved by ERMLE. The proposed methodology is further illustrated using two progressively Type-II censored real datasets from engineering reliability and biomedical survival analysis. Model adequacy is assessed through goodness-of-fit statistics with corresponding p-values, bootstrap confidence intervals, and graphical comparisons. The simulation results show that entropy regularization substantially improves estimation accuracy for the Exponential, Weibull, and Gamma distributions, particularly under moderate and heavy censoring, whereas only negligible improvements are observed for the Lognormal distribution. In the engineering reliability application, the Weibull distribution provides the best overall fit, while the Weibull and Gamma models exhibit the most satisfactory performance for the bladder cancer remission data. Across both applications, ERMLE yields parameter estimates and fitted models that are highly consistent with those of the classical MLE while providing stable estimation under progressive censoring. Overall, the proposed framework demonstrates that the effectiveness of entropy regularization is distribution-dependent rather than universal and provides practical guidance for selecting suitable estimation strategies in reliability and survival analysis.