  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.
