Modeling and Fitting the Alpha-Power Transformed Extended Exponential Distribution for Competing Risks: Methods and Applications

Abstract

Survival analysis concerns the modeling of time-to-event data, especially when the event of interest can occur due to multiple, mutually exclusive causes. In such contexts, competing risks models are essential, whether in reliability engineering-where components may fail for different reasons-or in medical research-where patients may die from various causes. The Alpha-Power Transformed Extended Exponential (APTEE) distribution, defined through the transformation of the Extended Exponential distribution, offers a highly flexible framework for modeling competing risks. It encompasses several classical distributions as special cases and accommodates various hazard rate shapes including increasing, decreasing, constant, bathtub, and upside-down bathtub forms. In this study, we estimate the parameters of the APTEE model using multiple estimation techniques such as maximum likelihood, Kolmogorov-Smirnov, Anderson-Darling, and Cramer-von Mises methods. Furthermore, we introduce a modified chi-square goodness-of-fit statistic that recovers information typically lost through data grouping and enhances the model selection process. We also conduct comprehensive model diagnostics, including Cox-Snell residual analysis, Q-Q plots, and hazard rate comparisons, to evaluate the adequacy and robustness of the fitted model. The proposed methodology is validated through extensive simulation studies and an application to real-world medical data, demonstrating the practical effectiveness of the APTEE model in survival and reliability analysis.