We develop a Bayesian model–based approach to finite population estimation accounting for spatial dependence. Our innovation here is a framework that achieves inference for finite population quantities in spatial process settings. A key distinction from the small area estimation setting is that we analyze finite populations referenced by their geographic coordinates. Specifically, we consider a two‐stage sampling design in which the primary units are geographic regions, the secondary units are point‐referenced locations, and the measured values are assumed to be a partial realization of a spatial process. Estimation of finite population quantities from geostatistical models does not account for sampling designs, which can impair inferential performance, whereas design‐based estimates ignore the spatial dependence in the finite population. We demonstrate by using simulation experiments that process‐based finite population sampling models improve model fit and inference over models that fail to account for spatial correlation. Furthermore, the process‐based models offer richer inference with spatially interpolated maps over the entire region. We reinforce these improvements and demonstrate scalable inference for groundwater nitrate levels in the population of California Central Valley wells by offering estimates of mean nitrate levels and their spatially interpolated maps.

中文翻译：

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空间过程设置下有限种群的贝叶斯推断

我们开发了一种基于贝叶斯模型的有限人口估计方法，该方法考虑了空间依赖性。我们在这里的创新是一个框架，可以在空间过程设置中推断出有限的种群数量。与小面积估算设置的主要区别在于，我们分析了由其地理坐标引用的有限人口。具体来说，我们考虑一个两阶段抽样设计，其中主要单位是地理区域，次要单位是点参考的位置，并且假定测量值是空间过程的部分实现。从地统计模型估计有限人口数量并未考虑抽样设计，这可能会削弱推理性能，而基于设计的估计会忽略有限人口中的空间依赖性。通过使用模拟实验，我们证明了基于过程的有限总体抽样模型可以改善模型拟合度和推理能力，而这些模型不能解决空间相关性问题。此外，基于过程的模型通过整个区域的空间插值图提供了更丰富的推断。我们加强了这些改进，并通过提供平均硝酸盐含量的估算值及其空间内插图，证明了加利福尼亚中央山谷水井中地下水硝酸盐含量的可扩展推断。