Functional central limit theorems for single-stage samplings designs

TL;DR

This paper establishes functional central limit theorems for empirical processes in survey sampling, providing theoretical foundations for joint model-based and design-based inference in single-stage unequal probability sampling designs.

Contribution

It introduces new functional CLTs for the Horvitz-Thompson and He1jek empirical processes under minimal correlation conditions, applicable to practical survey sampling.

Findings

Theorems hold under conditions on higher order correlations.

Results apply to both finite population and super-population means.

Simulation illustrates the limit behavior of a Hadamard differentiable functional.

Abstract

For a joint model-based and design-based inference, we establish functional central limit theorems for the Horvitz-Thompson empirical process and the H\'ajek empirical process centered by their finite population mean as well as by their super-population mean in a survey sampling framework. The results apply to single-stage unequal probability sampling designs and essentially only require conditions on higher order correlations. We apply our main results to a Hadamard differentiable statistical functional and illustrate its limit behavior by means of a computer simulation.