Martingale central-limit theorems for pivotal sampling

TL;DR

This paper proves the asymptotic normality of the Horvitz-Thompson estimator in ordered pivotal sampling using martingale central-limit theorems, accommodating spatial correlations for survey applications.

Contribution

It introduces a martingale CLT approach to establish asymptotic normality under both design-based and model-assisted frameworks, including correlated spatial data.

Findings

Asymptotic normality of the estimator is proven.

Applicable to spatial sampling with correlated data.

Supports both design-based and model-assisted approaches.

Abstract

Ordered pivotal sampling is one of the simplest algorithm to perform without-replacement unequal probability sampling. It has found uses in the context of longitudinal surveys and spatial sampling, and enables in particular a good spatial balance of the selected units. In this work, we follow the approach proposed by Ohlsson~(1986), and apply a martingale central-limit theorem to prove the asymptotic normality of the Horvitz-Thompson estimator under a design-based approach, and under a model-assisted approach. In particular, our model assumptions allow for correlations between values, which is of particular interest for applications in spatial sampling.