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. However, for practical purposes, 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：为估计“免疫”或“治愈”对象（即永远不会经历失败事件）的比例，需要足够长的随访期。文献中已提出若干统计检验来评估充分随访的假设，即研究时长应长于未治愈对象生存时间的支撑区间。然而，从实际应用角度而言，若研究结束后事件发生的概率极小，则随访可视为足够长。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，通过分布分位数来刻画，并开发了一种新型非参数统计检验方法。所提方法主要依赖分布尾部密度函数非递增的假设。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），并通过自助法计算临界值。通过广泛的模拟研究评估了该检验的性能，并将该方法应用于乳腺癌数据的实例分析。