On progressive first-failure reliability analysis: Classical and Bayesian approaches

The progressive first-failure censoring (PF-FC) plan is widely used in reliability settings where test items are arranged into groups of size $k$, and only the earliest failure in each group is observed. In this study, statistical inference for the Gompertz–Lindley distribution (GLD) under PF-FC was considered, with emphasis on estimating the model parameters together with the reliability and hazard rate functions (HRFs). Classical inference was performed via the maximum likelihood method (MLE), and confidence intervals (CIs) were formed using the large-sample behavior of the estimators. A Bayesian framework was also constructed using independent gamma priors and non-informative priors (NIPs) under loss structures. Markov Chain Monte Carlo (MCMC) algorithms were used to generate Bayesian estimates (BEs) and credible intervals (CRIs). Reliability measures are examined from both classical and BE. To evaluate the proposed procedures, a MCMC simulation study was carried out to examine their precision and robustness. The practical relevance of the developed methodology was illustrated using a real lifetime dataset.