Approximation of rejective sampling inclusion probabilities and application to high order correlations

Helène Boistard,Anne Ruiz-Gazen,Hendrik P Lopuhaä

doi:10.1214/12-ejs736

Abstract

This paper is devoted to rejective sampling. We provide an expansion of joint inclusion probabilities of any order in terms of the inclusion probabilities of order one, extending previous results by Hajek (1964) and Hajek (1981) and making the remainder term more precise. Following Hajek (1981), the proof is based on Edgeworth expansions. The main result is applied to derive bounds on higher order correlations, which are needed for the consistency and asymptotic normality of several complex estimators.

Highlights

In a finite population of size N, sampling without replacement with unequal inclusion probabilities and fixed sample size is not straightforward, but there exist several sampling designs that satisfy these properties (see Brewer and Hanif (1983) for a review)
Rejective sampling with size n can be regarded as Poisson sampling conditionally on the sample size being equal to n
The unconditional Poisson design can be implemented by drawing N independently distributed Bernoulli random variables with different probabilities of success, but it has the disadvantage of working with a random sample size

Summary

Introduction

In a finite population of size N , sampling without replacement with unequal inclusion probabilities and fixed sample size is not straightforward, but there exist several sampling designs that satisfy these properties (see Brewer and Hanif (1983) for a review). One interesting application of our result is that it enables us to show that rejective sampling satisfies the assumptions needed for the consistency and the asymptotic normality of some complex estimators, such as the ones defined in Breidt and Opsomer (2000), Breidt et al (2007), Cardot et al (2010) or Wang (2009) Such assumptions involve conditions on correlations up to order four, which are difficult to check for complex sampling designs that go beyond simple random sampling without replacement or Poisson sampling. In the case-control context, Arratia, Goldstein and Langholz (2005) consider rejective sampling and give approximations of higher order correlations Their approach and the assumptions they need to derive their results are different from the ones we consider in the present paper.

Notations and main result

Application

Proof of Lemma 1

Proofs of Lemma 2 and Proposition 1