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.

The competing risk (CR) model is crucial for studying various areas, such as biology, econometrics, and engineering. When multiple factors could cause a product to fail, these factors often work against each other, resulting in the product's failure. This scenario is known as the CR problem. This study focused on parameter estimation of the generalized Lomax distribution under a generalized progressive hybrid censoring scheme in the presence of CR when the cause of failure for each item was known and independent. Both maximum likelihood (ML) and Bayesian approaches were used to estimate the unknown parameters, reliability characteristics, and relative risks due to two causes. Bayesian estimators under gamma priors with different loss functions were generated using Markov chain Monte Carlo, and confidence intervals (CIs) were generated using the ML estimation method. Additionally, two bootstrap CIs for the unknown parameters were presented. According to the conditional posterior distribution, credible intervals and the highest posterior density intervals were further generated. The performance of different estimators was compared using Monte Carlo simulation, and real-data applications were used to verify the proposed estimates.