We consider nonparametric estimation of a distribution function when data are collected from two-phase stratified sampling without replacement. We study the inverse probability weighted empirical distribution function and propose a novel computational procedure to construct a confidence band. Two-phase sampling design induces heterogeneity across strata and dependence due to sampling without replacement. Two major statistical challenges from this design are: (1) the standard practice to approximate sampling without replacement by Bernoulli sampling leads to an incorrect coverage probability, and (2) a complicated limiting process of the proposed estimator does not allow one to analytically compute quantiles of the supremum of the limiting process nor to apply existing bootstrap methods to the proposed estimator. To address these issues, we rigorously establish the asymptotic properties of the proposed estimator and develop a simulation-based method to estimate the limiting process. The finite sample performance is evaluated through a simulation study. A Wilms tumor example is provided.

当从两阶段分层抽样中收集数据而不进行替换时，我们考虑分布函数的非参数估计。我们研究了逆概率加权经验分布函数，并提出了一种新的计算程序来构建置信带。两阶段采样设计会导致层间异质性，并且由于无需更换就可以进行采样。此设计面临的两个主要统计挑战是：（1）在不被伯努利抽样代替的情况下近似抽样的标准做法会导致覆盖概率不正确；（2）拟议估算器的复杂限制过程不允许人们分析计算分位数限制过程的最高点，也没有将现有的引导方法应用于拟议的估计器。为了解决这些问题，我们严格地建立了所提出的估计器的渐近性质，并开发了一种基于仿真的方法来估计极限过程。有限样本性能通过仿真研究进行评估。提供了威尔姆斯肿瘤的例子。