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

TL;DR

This paper develops an expansion for joint inclusion probabilities in rejective sampling, improving precision and enabling better bounds on higher order correlations crucial for estimator consistency and asymptotic normality.

Contribution

It extends previous results by providing a more precise expansion of inclusion probabilities of any order using Edgeworth expansions.

Findings

Derived bounds on higher order correlations

Extended He1jek's previous results

Improved understanding of estimator properties

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 H\'ajek (1964) and H\'ajek (1981) and making the remainder term more precise. Following H\'ajek (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.