Exponential inequalities for sampling designs

In this work, we introduce an approach based on the martingale 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. We derive a new exponential inequality for conditionally negatively associated (CNA) sampling designs, which is shown to improve over two existing inequalities that can be used in this context. We establish that Chao’s procedure, Tillé’s elimination procedure and the generalized Midzuno method are CNA sampling designs, and thus obtain an exponential inequality for these three sampling procedures. We show that our approach is useful beyond CNA sampling designs by deriving an exponential inequality for Brewer’s method.