Small area estimation is concerned with methodology for estimating population parameters associated with a geographic area defined by a cross‐classification that may also include non‐geographic dimensions. In this paper, we develop constrained estimation methods for small area problems: those requiring smoothness with respect to similarity across areas, such as geographic proximity or clustering by covariates, and benchmarking constraints, requiring weighted means of estimates to agree across levels of aggregation. We develop methods for constrained estimation decision theoretically and discuss their geometric interpretation. The constrained estimators are the solutions to tractable optimisation problems and have closed‐form solutions. Mean squared errors of the constrained estimators are calculated via bootstrapping. Our approach assumes the Bayes estimator exists and is applicable to any proposed model. In addition, we give special cases of our techniques under certain distributional assumptions. We illustrate the proposed methodology using web‐scraped data on Berlin rents aggregated over areas to ensure privacy.

中文翻译：

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小面积估计的平滑和基准化

小面积估算与估算与交叉分类定义的地理区域相关的人口参数的方法有关，该交叉分类也可能包括非地理维度。在本文中，我们针对小面积问题开发了一种受限的估计方法：那些需要在区域间的相似度方面保持平滑的方法，例如地理接近度或通过协变量进行聚类，以及基准约束，需要加权的估计方法才能在聚合级别上达成共识。我们从理论上开发了约束估计决策的方法，并讨论了它们的几何解释。约束估计量是可解决的优化问题的解决方案，并且具有封闭形式的解决方案。约束估计量的均方误差通过自举计算。我们的方法假设存在贝叶斯估计量，并且适用于任何建议的模型。另外，我们在某些分布假设下给出了我们技术的特殊情况。我们使用网络收集的有关柏林租金汇总的数据来说明提议的方法，该数据汇总了各个地区的租金以确保隐私。