December 2015 Coupling methods for multistage sampling
Guillaume Chauvet
Ann. Statist. 43(6): 2484-2506 (December 2015). DOI: 10.1214/15-AOS1348

## 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. Kutató Int. Közl. 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.

## Citation

Guillaume Chauvet. "Coupling methods for multistage sampling." Ann. Statist. 43 (6) 2484 - 2506, December 2015. https://doi.org/10.1214/15-AOS1348

## Information

Received: 1 May 2015; Revised: 1 May 2015; Published: December 2015
First available in Project Euclid: 7 October 2015

zbMATH: 1331.62071
MathSciNet: MR3405601
Digital Object Identifier: 10.1214/15-AOS1348

Subjects:
Primary: 62D05
Secondary: 62E20 , 62G09

Keywords: bootstrap , coupling algorithm , without-replacement sampling , with-replacement sampling