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.

中文翻译：

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具有繁琐泛函和复杂调查数据的增强两步估计方程

在复杂调查数据中存在滋扰泛函的情况下进行统计推断是社会和经济研究中的一个重要课题。基尼指数、洛伦兹曲线和分位数份额是常见的例子。讨厌的泛函通常由插件非参数估计器处理，主要的推理过程可以通过两步广义经验似然法来执行。不幸的是，由此产生的推断效率不高，即使在简单的随机抽样下，威尔克斯定理的非参数版本也会崩溃。我们提出了一种具有繁琐泛函和复杂调查的增强估计方程方法。第二步增强估计函数遵循奈曼正交条件并自动处理第一步插件估计器的影响，并且所得到的主要感兴趣参数的估计量对于第一步方法是不变的。更重要的是，基于威尔克斯定理的广义经验似然对于常用调查设计的基于设计的框架下的主要感兴趣参数成立，并且最大广义经验似然估计达到了半参数效率界限。通过模拟研究和使用纽约市社会指标调查数据集的应用程序证明了所提出方法的性能。最大广义经验似然估计达到半参数效率界限。通过模拟研究和使用纽约市社会指标调查数据集的应用程序证明了所提出方法的性能。最大广义经验似然估计达到半参数效率界限。通过模拟研究和使用纽约市社会指标调查数据集的应用程序证明了所提出方法的性能。