Abstract
The adaptive progressive first‐failure censoring scheme is a flexible life‐testing design that integrates the efficiency of first‐failure experiments with the adaptability of progressive censoring, ensuring the observation of a predetermined number of failures while allowing dynamic control over test duration and resource allocation. Despite its practical advantages in reliability experimentation, statistical inference under this censoring structure remains limited in the existing literature, particularly with respect to the estimation of reliability measures. Motivated by this gap, this paper develops classical and Bayesian inference for the XLindley lifetime distribution under the adaptive progressive first‐failure censoring scheme. The XLindley distribution, characterized by a single positive parameter, provides sufficient flexibility to model positively skewed lifetime data while maintaining analytical tractability under complex censoring mechanisms. Maximum likelihood estimation is derived for the model parameter as well as for the reliability and hazard rate functions, and two types of approximate confidence intervals are constructed based on normal approximation and log‐transformed estimators. Bayesian estimation is implemented under the squared error loss function using Markov chain Monte Carlo simulation, from which symmetric Bayesian credible intervals and highest posterior density intervals are obtained. The finite‐sample behavior of the proposed estimators is examined through an extensive Monte Carlo simulation study. The methodology is further illustrated through the analysis of two real datasets, including survival times of gastric cancer patients receiving combined chemotherapy and radiation therapy, and monthly rainfall measurements recorded in New South Wales, Australia. The results demonstrate the flexibility of the XLindley model and the effectiveness of the proposed inferential procedures under adaptive progressive first‐failure censoring.