Asymptotic properties of pivotal sampling with application to spatial sampling

Unequal probability sampling without replacement is commonly used for sample selection. To produce estimators with associated condence intervals, some basic statistical properties like consistency and asymptotic normality of the Horvitz-Thompson estimator are desirable. These properties have been mainly studied for large entropy sampling designs. On the other hand, spatial sampling designs rather make use of sampling algorithms which take into account the order of units in the population, like systematic sampling or pivotal sampling. So far, the statistical properties of such procedures have not been investigated. In this work, we study the asymptotic properties of the pivotal sampling design. Under mild assumptions, we prove that the Horvitz-Thompson estimator is asymptotically normally distributed and that a conservative variance estimator can always be computed. We also introduce a general spatial sampling design which is spatially balanced, which possesses good statistical properties and which is computationally very ecient, even for large databases