Inference for two-stage sampling designs with application to a panel for urban policy

TL;DR

This paper establishes the statistical properties of the Horvitz-Thompson estimator in two-stage sampling designs, including consistency and asymptotic normality, with practical applications to urban policy surveys.

Contribution

It extends the theoretical understanding of two-stage sampling estimators, proving their consistency and normality under mild conditions, and introduces simplified variance estimators for large entropy designs.

Findings

Proved consistency of the Horvitz-Thompson estimator in two-stage designs.

Established asymptotic normality for large entropy first-stage sampling.

Proposed simplified variance estimators for negligible first-stage sampling fractions.

Abstract

Two-stage sampling designs are commonly used for household and health surveys. To produce reliable estimators with assorted confidence intervals, some basic statistical properties like consistency and asymptotic normality of the Horvitz-Thompson estimator are desirable, along with the consistency of assorted variance estimators. These properties have been mainly studied for single-stage sampling designs. In this work, we prove the consistency of the Horvitz-Thompson estimator and of associated variance estimators for a general class of two-stage sampling designs, under mild assumptions. We also study two-stage sampling with a large entropy sampling design at the first stage, and prove that the Horvitz-Thompson estimator is asymptotically normally distributed through a coupling argument. When the first-stage sampling fraction is negligible, simplified variance estimators which do not require estimating the variance within the Primary Sampling Units are proposed, and shown to be consistent. An application to a panel for urban policy, which is the initial motivation for this work, is also presented.