Augmented two-step estimating equations with nuisance functionals and complex survey data.

This article focuses on developing an augmented two-step generalized empirical likelihood (GEL) method for statistical inference on finite population parameters defined through estimating equations with nuisance functionals in complex survey data. The proposed approach introduces augmentation terms that satisfy the Neyman orthogonality condition, rendering the estimator of the main parameters invariant to the first-step plug-in estimator of the nuisance functional and achieving the semiparametric efficiency bound under a design-based framework. The method restores a nonparametric version of Wilks' theorem for the GEL ratio statistic under commonly used survey designs, facilitating valid confidence regions and hypothesis tests without complicated variance estimation. Applications to income inequality measures such as the Gini coefficient, Lorenz curves, and quantile shares are illustrated, with simulation studies and an analysis of the New York City Social Indicators Survey demonstrating improved efficiency and coverage properties compared to conventional two-step methods.