Reliability estimation using expected and hierarchical Bayesian approach under progressive censoring with engineering application

In this present work, we propose the expected Bayesian (E- Bayesian) and hierarchical Bayesian (H- Bayesian) approaches to estimate the shape parameter and hazard rate under progressive Type-II censoring scheme. These estimates have been obtained using gamma priors based on various loss functions such as squared error, entropy, weighted balance, and minimum expected loss functions. Properties of E- and H- Bayesian estimates have been discussed. Further the relations between E- and H- Bayesian estimates based on different loss functions are provided. An investigation is carried out using Monte Carlo simulation to evaluate the effectiveness of the suggested estimators. The simulation provides a quantitative assessment of the estimates' accuracy and efficiency under various conditions by comparing them in terms of mean squared error. Additionally, the failure data set for ball bearings is examined to offer real-world examples of how the suggested estimations may be used and performed.