In order to estimate the proportion of `immune' or `cured' subjects who will never experience failure, a sufficiently long follow-up period is required. Several statistical tests have been proposed in the literature for assessing the assumption of sufficient follow-up, meaning that the study duration is longer than the support of the survival times for the uncured subjects. These tests do not perform satisfactorily, especially in terms of Type I error. In addition, they are constructed based on the assumption that the survival time for the uncured subjects has a compact support, i.e. the existence of a `cure time'. However, for practical purposes, the assumption of `cure time' is not realistic and the follow-up would be considered sufficiently long if the probability for the event to happen after the end of the study is very small. Based on this observation, we formulate a more relaxed notion of `practically' sufficient follow-up characterized by the quantiles of the distribution and develop a novel nonparametric statistical test. The proposed method relies mainly on the assumption of a non-increasing density function in the tail of the distribution. The test is then based on a shape constrained density estimator such as the Grenander or the kernel smoothed Grenander estimator and a bootstrap procedure is used for computation of the critical values. The performance of the test is investigated through an extensive simulation study, and the method is illustrated on breast cancer data.

翻译：为了估计那些永远不会经历失败的“免疫”或“治愈”受试者的比例，需要一个足够长的随访期。文献中已提出了几种统计检验方法，用于评估充分随访的假设，即研究持续时间长于未治愈受试者生存时间的支撑集。这些检验的表现并不令人满意，特别是在第一类错误方面。此外，这些检验的构建基于未治愈受试者的生存时间具有紧支撑的假设，即存在一个“治愈时间”。然而，出于实际目的，“治愈时间”的假设并不现实，如果研究结束后事件发生的概率非常小，随访即可被认为是充分长的。基于这一观察，我们提出了一个更宽松的“实际”充分随访概念，该概念由分布的分位数表征，并开发了一种新的非参数统计检验方法。所提出的方法主要依赖于分布尾部密度函数非增的假设。该检验基于形状约束密度估计量，如 Grenander 估计量或核平滑 Grenander 估计量，并使用自助法计算临界值。通过广泛的模拟研究检验了该检验的性能，并在乳腺癌数据上演示了该方法。