Asymptotic Normality under Two-Phase Sampling Designs

Large sample properties of statistical inferences in the context of finite populations are harder to determine than in the iid case due to their dependence jointly on the characteristics of the finite population and the sampling design employed. There have been many discussions on special inference procedures under special sampling designs in the literature. General and comprehensive results are still lacking. In this paper, we first present a surprising result on the weak law of large numbers under simple random sampling design. We show that the sampling mean is not necessarily consistent for the population mean even if the population first absolute moment is bounded by a constant not depending on the evolving population size. Instead, a sufficient condition requires the boundedness of the (1 + δ)th absolute population moment for some δ > 0. Based on this result, we prove asymptotic normality of a class of estimators under two-phase sampling design. We show that these estimators can typically be decomposed as sum of two random variables such that the first one is conditionally asymptotically normal and the second one is asymptotically normal. A theoretical result is derived to combine these two conclusions to prove the asymptotic normality of the estimators.