Nonparametric methods for comparing distribution functionals for dependent samples with application to inequality measures

This paper proposes asymptotically distribution-free inference methods for comparing a broad range of welfare indices across dependent samples, including those employed in inequality, poverty, and risk analysis. Two distinct situations are considered. \emph{First}, we propose asymptotic and bootstrap intersection methods which are completely robust to arbitrary dependence between two samples. \emph{Second}, we focus on the common case of overlapping samples -- a special form of dependent samples where sample dependence arises solely from matched pairs -- and provide asymptotic and bootstrap methods for comparing indices. We derive consistent estimates for asymptotic variances using the influence function approach. The performance of the proposed methods is studied in a simulation experiment: we find that confidence intervals with overlapping samples exhibit satisfactory coverage rates with reasonable precision, whereas conventional methods based on an assumption of independent samples have an inferior performance in terms of coverage rates and interval widths. Asymptotic inference can be less reliable when dealing with heavy-tailed distributions, while the bootstrap method provides a viable remedy, unless the variance is substantial or nonexistent. The intersection method yields reliable results with arbitrary dependent samples, including instances where overlapping samples are not feasible. We demonstrate the practical applicability of our proposed methods in analyzing dynamic changes in household financial inequality in Italy over time.