This paper is devoted to rejective sampling. We provide an expansion of joint inclusion probabilities of any order in terms of the inclusion probabilities of order one, extending previous results by Hajek (1964) and Hajek (1981) and making the remain
der term more precise. Following Hajek (1981), the proof is based on Edgeworth expansions. The main result is applied to derive bounds on higher order correlations, which are needed for the consistency and asymptotic normality of several complex estimators.
Currently, the high-precision estimation of nonlinear parameters such as Gini indices, low-income proportions or other measures of inequality is particularly crucial. In the present paper, we propose a general class of estimators for such parameters
that take into account univariate auxiliary information assumed to be known for every unit in the population. Through a nonparametric model-assisted approach, we construct a unique system of survey weights that can be used to estimate any nonlinear parameter associated with any study variable of the survey, using a plug-in principle. Based on a rigorous functional approach and a linearization principle, the asymptotic variance of the proposed estimators is derived, and variance estimators are shown to be consistent under mild assumptions. The theory is fully detailed for penalized B-spline estimators together with suggestions for practical implementation and guidelines for choosing the smoothing parameters. The validity of the method is demonstrated on data extracted from the French Labor Force Survey. Point and confidence intervals estimation for the Gini index and the low-income proportion are derived. Theoretical and empirical results highlight our interest in using a nonparametric approach versus a parametric one when estimating nonlinear parameters in the presence of auxiliary information.
