Nonparametric inference for distribution functions with stratified samples

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.