Estimation of the cure rate for distributions in the Gumbel maximum domain of attraction under insufficient follow-up[Cure models in survival analysis]

SummaryEstimators of the cured proportion from survival data which may include observations on cured subjects can only be expected to perform well when the follow-up period is sufficient. When follow-up is not sufficient, and the survival distribution of those susceptible to the event belongs to the Fréchet maximum domain of attraction, a nonparametric estimator for the cure proportion proposed by Escobar-Bach & Van Keilegom (2019) incorporates an adjustment that reduces the bias in the usual estimator. Besides the Fréchet, an important class of limiting distributions for maxima is the Gumbel class. We show that a very wide class of commonly used survival distributions, the generalized Gamma distributions, are in the Gumbel domain of attraction. Extrapolation techniques from extreme value theory are then used to derive, for distributions in this class, a nonparametric estimator of the cure proportion that is consistent and asymptotically normally distributed under reasonable assumptions, and performs well in simulation studies with data where follow-up is insufficient. We illustrate its use with an application to survival data where patients with differing stages of breast cancer have varying degrees of follow-up.