Two‐stage sampling designs are commonly used for household and health surveys. To produce reliable estimators with associated confidence intervals, some basic statistical properties like consistency and asymptotic normality of the Horvitz–Thompson estimator are desirable, along with the consistency of associated 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.

中文翻译：

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两阶段采样设计的推论

两阶段抽样设计通常用于家庭和健康调查。为了产生具有相关置信区间的可靠估计量，需要一些基本的统计属性，例如Horvitz-Thompson估计量的一致性和渐近正态性，以及相关的方差估计量的一致性。这些特性主要用于单阶段采样设计。在这项工作中，我们在温和的假设下证明了一般两阶段抽样设计类别的Horvitz-Thompson估计量和相关方差估计量的一致性。我们还在第一阶段研究了采用大熵抽样设计的两阶段抽样，并证明了霍维兹-汤普森估计量通过耦合参数呈渐近正态分布。如果第一阶段的采样比例可以忽略不计，提出了简化的方差估计器，该估计器不需要估计主要采样单位内的方差，并且证明是一致的。还介绍了城市政策专家组的申请，这是这项工作的最初动机。