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Local polynomial estimation for a small area mean under informative sampling

机译：在信息采样下，小区的局部多项式估计意味着

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Model-based methods are required to estimate small area parameters of interest, such as totals and means, when traditional direct estimation methods cannot provide adequate precision. Unit level and area level models are the most commonly used ones in practice. In the case of the unit level model, efficient model-based estimators can be obtained if the sample design is such that the sample and population models coincide: that is, the sampling design is non-informative for the model. If on the other hand, the sampling design is informative for the model, the selection probabilities will be related to the variable of interest, even after conditioning on the available auxiliary data. This will imply that the population model no longer holds for the sample. Pfeffermann and Sverchkov (2007) used the relationships between the population and sample distribution of the study variable to obtain approximately unbiased semi-parametric predictors of the area means under informative sampling schemes. Their procedure is valid for both sampled and non-sampled areas. Verret, Rao and Hidiroglou (2015) studied alternative procedures that incorporate a suitable function of the unit selection probabilities as an additional auxiliary variable. Their procedure resulted in approximately unbiased empirical best linear unbiased prediction (EBLUP) estimators for the small area means. In this paper, we extend the Verret et al. (2015) procedure by not assuming anything about the inclusion probabilities. Rather, we incorporate them into the unit level model via a smooth function of the inclusion probabilities. This function is estimated via a local approximation resulting in a local polynomial estimator. A conditional bootstrap method is proposed for the estimation of mean squared error (MSE) of the local polynomial and EBLUP estimators. The bias and efficiency properties of the local polynomial estimator are investigated via a simulation. Results for the bootstrap estimator of MSE are also presented.
机译：基于模型的方法需要估计兴趣的小区域参数，例如总计和手段，当传统的直接估计方法不能提供足够的精度时。单位等级和面积级模型是最常用的实践中使用的模型。在单位模型的情况下，如果样本设计使样品和群体模型重合，则可以获得有效的基于模型的估计器：即，采样设计对模型是非信息性的。另一方面，采样设计是对模型的信息性，选择概率将与感兴趣的变量有关，即使在可用辅助数据上调节后也会有关。这将暗示人口模型不再适用于样本。 Pfeffermann和Sverchkov（2007）使用了研究变量的人口和样本分布之间的关系，以在信息采样方案下获得区域意味着的大致无偏见的半参数预测器。它们的程序对于采样和非采样区域有效。 Verret，Rao和Hidiroglou（2015）研究了将单位选择概率的合适功能的替代程序作为额外的辅助变量。它们的程序导致小区域意味着大致无偏见的经验最佳线性无偏析预测（EBLUP）估计。在本文中，我们延伸了Verret等人。 （2015）除了任何关于包含概率的内容，程序的程序。相反，我们通过包含概率的平滑函数将它们纳入单位级模型。通过局部近似估计该函数，导致局部多项式估计器。提出了一种用于估计本地多项式和EBLUP估计器的平均平方误差（MSE）的条件引导方法。通过模拟研究了局部多项式估计器的偏差和效率特性。还提出了MSE的引导估计结果。

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