The Annals of Statistics

Coupling methods for multistage sampling

Guillaume Chauvet

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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.

Article information

Ann. Statist., Volume 43, Number 6 (2015), 2484-2506.

Received: May 2015
Revised: May 2015
First available in Project Euclid: 7 October 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62D05: Sampling theory, sample surveys
Secondary: 62E20: Asymptotic distribution theory 62G09: Resampling methods

Bootstrap coupling algorithm with-replacement sampling without-replacement sampling


Chauvet, Guillaume. Coupling methods for multistage sampling. Ann. Statist. 43 (2015), no. 6, 2484--2506. doi:10.1214/15-AOS1348.

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Supplemental materials

  • Supplement to “Coupling methods for multistage sampling”. The supplement [7] contains additional proofs of Propositions in Section 1, and additional simulation results in Section 2.