Coupling methods for multistage sampling

TL;DR

This paper develops coupling methods to analyze multistage sampling designs, enabling the derivation of asymptotic properties and bootstrap consistency for complex survey sampling procedures.

Contribution

It generalizes existing coupling techniques to multistage sampling with various first-stage designs, establishing new central limit theorems and bootstrap consistency results.

Findings

Established a CLT for the Horvitz--Thompson estimator under multistage sampling.

Proved bootstrap consistency for simple random without replacement sampling at the first stage.

Extended coupling methods to more general multistage sampling schemes.

Abstract

Multistage sampling is commonly used for household surveys when there exists no sampling frame, or when the population is scattered over a wide area. Multistage sampling usually introduces a complex dependence in the selection of the final units, which makes asymptotic results quite difficult to prove. In this work, we consider multistage sampling with simple random without replacement sampling at the first stage, and with an arbitrary sampling design for further stages. We consider coupling methods to link this sampling design to sampling designs where the primary sampling units are selected independently. We first generalize a method introduced by [Magyar Tud. Akad. Mat. Kutat\'{o} Int. K\"{o}zl. 5 (1960) 361-374] to get a coupling with multistage sampling and Bernoulli sampling at the first stage, which leads to a central limit theorem for the Horvitz--Thompson estimator. We then introduce a new coupling method with multistage sampling and simple random with replacement sampling at the first stage. When the first-stage sampling fraction tends to zero, this method is used to prove consistency of a with-replacement bootstrap for simple random without replacement sampling at the first stage, and consistency of bootstrap variance estimators for smooth functions of totals.