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We develop large sample theory for merged data from multiple sources. Mainstatistical issues treated in this paper are (1) the same unit potentiallyappears in multiple datasets from overlapping data sources, (2) duplicateditems are not identified, and (3) a sample from the same data source isdependent due to sampling without replacement. We propose and study a newweighted empirical process and extend empirical process theory to a dependentand biased sample with duplication. Specifically, we establish the uniform lawof large numbers and uniform central limit theorem over a class of functionsalong with several empirical process results under conditions identical tothose in the i.i.d. setting. As applications, we study infinite-dimensionalM-estimation and develop its consistency, rates of convergence, and asymptoticnormality. Our theoretical results are illustrated with simulation studies anda real data example.

Large Sample Theory for Merged Data from Multiple Sources