Confidence bands for a distribution function with merged data from multiple sources

We consider nonparametric estimation of a distribution function when data are collected from multiple overlapping data sources. Main statistical challenges include (1) heterogeneity of data sets, (2) unidentified duplicated records across data sets, and (3) dependence due to sampling without replacement from a data source. The proposed estimator is computable without identifying duplication but corrects bias from duplicated records. We show the uniform consistency of the proposed estimator over the real line and its weak convergence to a Gaussian process. Based on these asymptotic properties, we propose a simulation-based confidence band that enjoys asymptotically correct coverage probability. The finite sample performance is evaluated through a simulation study. A Wilms tumor example is provided.