Exponential inequalities for sampling designs

TL;DR

This paper develops a unified approach using martingale representation and Azuma-Hoeffding's inequality to derive exponential bounds for sampling estimators, improving understanding of their concentration properties.

Contribution

It introduces a general method for exponential inequalities in sampling designs, applying it to several procedures and establishing their negative association properties.

Findings

Derived exponential inequalities for various sampling methods.

Proved negative association for Chao's, Tillé's, and Midzuno's sampling designs.

Showed the new inequalities are sharper than existing bounds for negatively associated variables.

Abstract

In this work we introduce a general approach, based on the mar-tingale representation of a sampling design and Azuma-Hoeffding's inequality , to derive exponential inequalities for the difference between a Horvitz-Thompson estimator and its expectation. Applying this idea, we establish such inequalities for Chao's procedure, Till{\'e}'s elimination procedure, the generalized Midzuno method as well as for Brewer's method. As a by-product, we prove that the first three sampling designs are (conditionally) negatively associated. For such sampling designs, we show that that the inequality we obtain is usually sharper than the one obtained by applying known results for negatively associated random variables.