## The Annals of Statistics

- Ann. Statist.
- Volume 45, Number 4 (2017), 1728-1758.

### Functional central limit theorems for single-stage sampling designs

Hélène Boistard, Hendrik P. Lopuhaä, and Anne Ruiz-Gazen

#### Abstract

For a joint model-based and design-based inference, we establish functional central limit theorems for the Horvitz–Thompson empirical process and the Hájek 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.

#### Article information

**Source**

Ann. Statist., Volume 45, Number 4 (2017), 1728-1758.

**Dates**

Received: September 2015

Revised: August 2016

First available in Project Euclid: 28 June 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1498636872

**Digital Object Identifier**

doi:10.1214/16-AOS1507

**Mathematical Reviews number (MathSciNet)**

MR3670194

**Zentralblatt MATH identifier**

06773289

**Subjects**

Primary: 62D05: Sampling theory, sample surveys

**Keywords**

Design and model-based inference Hájek Process Horvitz–Thompson process rejective sampling Poisson sampling high entropy designs poverty rate

#### Citation

Boistard, Hélène; Lopuhaä, Hendrik P.; Ruiz-Gazen, Anne. Functional central limit theorems for single-stage sampling designs. Ann. Statist. 45 (2017), no. 4, 1728--1758. doi:10.1214/16-AOS1507. https://projecteuclid.org/euclid.aos/1498636872

#### Supplemental materials

- Supplement to “Functional central limit theorems for single-stage samplings designs.”. Appendix A: Proofs for results in the main text. This supplement contains detailed proofs of lemmas, propositions and corollaries for results in the main text that are not present in Section 9. Appendix B: Additional technicalities. This supplement contains detailed proofs of some remarks and additional lemmas. Appendix C: Fixed size sampling designs with deterministic inclusion probabilities. This supplement contains results for fixed size sampling designs with deterministic inclusion probabilities, obtained under alternative conditions for (C2)–(C4).Digital Object Identifier: doi:10.1214/16-AOS1507SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.