Large Sample Theory for Merged Data from Multiple Sources

TL;DR

This paper develops large sample statistical theory for merged data from multiple sources, addressing issues like duplication, dependence, and bias, and extends empirical process theory to such complex data structures.

Contribution

It introduces a new weighted empirical process framework and extends empirical process theory to dependent, biased, and duplicated data, enabling rigorous statistical inference.

Findings

Established uniform law of large numbers and central limit theorem for complex data

Proved consistency, convergence rates, and asymptotic normality for infinite-dimensional M-estimators

Validated theoretical results through simulations and real data analysis

Abstract

We develop large sample theory for merged data from multiple sources. Main statistical issues treated in this paper are (1) the same unit potentially appears in multiple datasets from overlapping data sources, (2) duplicated items are not identified, and (3) a sample from the same data source is dependent due to sampling without replacement. We propose and study a new weighted empirical process and extend empirical process theory to a dependent and biased sample with duplication. Specifically, we establish the uniform law of large numbers and uniform central limit theorem over a class of functions along with several empirical process results under conditions identical to those in the i.i.d. setting. As applications, we study infinite-dimensional M-estimation and develop its consistency, rates of convergence, and asymptotic normality. Our theoretical results are illustrated with simulation studies and a real data example.