Advancing Competing Risks Analysis via Adaptive Progressive First-Failure Censoring: The Xgamma Distribution Case with an Application

The analysis of lifetime data in the presence of multiple competing causes of failure remains a fundamental problem in reliability and survival studies, particularly when observations are obtained under complex and resource-constrained censoring mechanisms. In this context, the adaptive progressive first-failure censoring scheme has emerged as a flexible design that allows the removal mechanism to adapt dynamically to the observed evolution of failures, making it well suited for modern life-testing experiments involving grouped units and limited resources. This paper develops statistical inference for the Xgamma competing risks model under the adaptive progressive first-failure censoring scheme. Both classical and Bayesian approaches are employed for parameter estimation and reliability analysis. Maximum likelihood estimation is carried out despite the inherent analytical complexity of the resulting nonlinear likelihood function, and two types of approximate confidence intervals are constructed based on direct and transformation-based normal approximations. From a Bayesian perspective, independent gamma prior distributions are assumed, and posterior inference is performed using Markov chain Monte Carlo methods to address the intractability of the posterior distribution. Bayesian point estimators, along with symmetric credible intervals and highest posterior density intervals, are obtained from the generated posterior samples. The proposed methods are evaluated through an extensive simulation study conducted under a variety of experimental settings. In addition, the practical applicability of the developed methodology is illustrated through the analysis of real data involving relapse times of multiple myeloma patients. The results demonstrate that the proposed approach yields accurate estimation and reliable uncertainty quantification, confirming its effectiveness for analyzing competing risks data under advanced censoring schemes.