  Statistical inference in the presence of nuisance functionals with complex
survey data is an important topic in social and economic studies. The Gini
index, Lorenz curves and quantile shares are among the commonly encountered
examples. The nuisance functionals are usually handled by a plug-in
nonparametric estimator and the main inferential procedure can be carried out
through a two-step generalized empirical likelihood method. Unfortunately, the
resulting inference is not efficient and the nonparametric version of the
Wilks' theorem breaks down even under simple random sampling. We propose an
augmented estimating equations method with nuisance functionals and complex
surveys. The second-step augmented estimating functions obey the Neyman
orthogonality condition and automatically handle the impact of the first-step
plug-in estimator, and the resulting estimator of the main parameters of
interest is invariant to the first step method. More importantly, the
generalized empirical likelihood based Wilks' theorem holds for the main
parameters of interest under the design-based framework for commonly used
survey designs, and the maximum generalized empirical likelihood estimators
achieve the semiparametric efficiency bound. Performances of the proposed
methods are demonstrated through simulation studies and an application using
the dataset from the New York City Social Indicators Survey.
