Defective Gamma–G Family for Cure Fraction Models: Novel Survival Methods with Applications to Cancer Data

In this paper, we propose two novel defective survival models within the Gamma–G family: the defective Gamma–Gompertz and the defective Gamma–Dagum distributions. In contrast to the corresponding Gamma–G mixture cure formulation, in which the Gamma–G distributional parameters are combined with an explicit cure fraction mixing parameter, the proposed defective formulation induces the cure fraction through the limiting behavior of the survival function. Thus, within the same Gamma–G baseline structure, the model avoids introducing an additional cure fraction parameter. The motivation for these new models lies in the limited set of defective distributions currently available, despite the increasing demand for flexible cure rate models in biomedical applications. By extending the defective property to the Gamma–G construction, our approach fills this methodological gap while providing models that are both interpretable and computationally efficient. We show that the Gamma–G construction preserves defectiveness whenever the baseline distribution is defective, thus establishing a coherent theoretical foundation. Both models allow covariate effects through regression structures on shape, scale, and, in the case of the Gamma–Dagum distribution, on the cure fraction parameter, resulting in flexible and interpretable specifications. Parameters are estimated via maximum likelihood, and an extensive Monte Carlo study confirms estimator consistency and accurate coverage in finite samples. The practical relevance of the models is illustrated with two large clinical datasets on melanoma and cervical cancer from the São Paulo Cancer Registry. Results reveal that the proposed models provide competitive goodness-of-fit and offer useful insights into long-term survival compared to traditional cure rate approaches. Overall, this work introduces a unifying and flexible framework for defective survival models, extending their applicability and delivering practical improvements over existing cure models.