Optimal Unified Progressive Hybrid Censoring for Log‐Logistic Reliability Models: Applications in Engineering, Clinical, and Chemical Laboratories

ABSTRACT This study develops an integrated inferential and design‐optimal framework for lifetime analysis under a unified progressive hybrid log‐logistic censoring scheme, motivated by the practical demands of reliability experiments involving rare and high‐value materials. By unifying progressive removal mechanisms with hybrid time–failure termination rules, the proposed censoring structure achieves a flexible balance between experimental cost, duration control, and statistical efficiency. Within this framework, we investigate both frequentist and Bayesian inference for the log‐logistic lifetime model, deriving maximum likelihood estimators for model parameters and associated reliability characteristics, including survival and hazard rate functions. Asymptotic confidence intervals are constructed using Fisher information and delta‐method approximations, while Bayesian inference is performed via Markov chain Monte Carlo methods under independent gamma priors, with posterior computation implemented through a Metropolis–Hastings algorithm to address analytical intractability. Beyond estimation, the study advances optimum progressive censoring design by systematically comparing competing censoring plans using multiple optimality criteria, thereby identifying strategies that maximize inferential precision under practical constraints. Extensive Monte Carlo simulations demonstrate that Bayesian estimators, particularly under informative priors, consistently outperform their frequentist counterparts in terms of bias reduction, mean squared error, and interval efficiency, especially in heavily censored or moderate‐sample scenarios. The proposed methodology is further illustrated through real datasets from diamond durability and gold reliability studies, highlighting its effectiveness in modeling rare‐event behavior and complex hazard structures in materials science. Overall, this work contributes a unified and practically implementable framework that couples advanced censoring design with robust inferential tools, offering significant gains in efficiency and interpretability for modern reliability and survival analysis.