Efficient estimation of non-linear finite population parameters by using non-parametrics

Currently, high precision estimation of non-linear parameters such as Gini indices, low income proportions or other measures of inequality is particularly crucial. 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 non-parametric model-assisted approach, we construct a unique system of survey weights that can be used to estimate any non-linear parameter that is 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 estimators proposed 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 Labour Force Survey. Point and confidence interval estimation for the Gini index and the low income proportion are derived. Theoretical and empirical results highlight our interest in using a non-parametric versus a parametric approach when estimating non-linear parameters in the presence of auxiliary information.