Analytic inference in finite population framework via resampling

The aim of this dissertation is to provide nonparametric tools for analytic inference on superpopulation models. To pursue the goal we approach to the problem in two different ways. The first one is analytic. Following the classical empirical process theory, we first derive a functional central limit theorem that fully characterizes the asymptotic distribution of the Hàjek estimator of the distribution function of the superpopulation. In addition, assuming some regularity conditions on the (superpopulation) parameters of interest, we extend this analytic characterization to a large class of possible paramaters of interest. The second one is more “practical”: our aim is to construct a computer intensive procedure that allows to infer the superpopulation, also when the (asymptotic) distribution of an interest parameter has an unmanageable analytic form. Clearly, such a procedure is resampling. Unfortunately, the most famous resampling technique, the bootstrap procedure, does not work in our framework. In fact, in the finite population framework, even if a superpopulation is assumed, the units cannot be assumed independent in the presence of a non trivial sampling design. This fact makes the classic bootstrap fail. Of course, in the survey sampling literature, resampling procedure have been proposed, but we haven not resort to them because of two reasons: i) a largest part of these resampling techniques have been developed to infer the finite population and not the superpopulation; ii) we want to make a parallel between the classical non parametric theory and survey sampling. Almost all of these procedures are justified by mimicking the first two moments of the distribution of the considered estimator, and this is not the argument used to justify Efron’s bootstrap in classical nonparametric statistics. Thus, we introduce the “ multinomial” scheme as a resampling procedure for the superpopulation and we provide an asymptotic validation of this method, that involves the whole distribution of the considered estimators, exactly as it happens for classic bootstrap. In the last part of this work, the results obtained are applied to different inferential problems and, for each one of the concerned problem, a simulation study is performed to test the validity of our proposal. For these applications, we especially focused on problems where the interest parameter is not a linear function of the data.