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Two-stage sampling designs are commonly used for household and healthsurveys. To produce reliable estimators with assorted confidence intervals,some basic statistical properties like consistency and asymptotic normality ofthe Horvitz-Thompson estimator are desirable, along with the consistency ofassorted variance estimators. These properties have been mainly studied forsingle-stage sampling designs. In this work, we prove the consistency of theHorvitz-Thompson estimator and of associated variance estimators for a generalclass of two-stage sampling designs, under mild assumptions. We also studytwo-stage sampling with a large entropy sampling design at the first stage, andprove that the Horvitz-Thompson estimator is asymptotically normallydistributed through a coupling argument. When the first-stage sampling fractionis negligible, simplified variance estimators which do not require estimatingthe variance within the Primary Sampling Units are proposed, and shown to beconsistent. An application to a panel for urban policy, which is the initialmotivation for this work, is also presented.

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