Bayesian Finite Population Modelingfor Spatial Process Settings

We develop a Bayesian model-based approach to finite population estimation accounting for spatialdependence. Our innovation here is a framework that achieves inference for finite population quantities inspatial process settings. A key distinction from the small area estimation setting is that we analyze finitepopulations referenced by their geographic coordinates (point-referenced data). Specifically, we consider atwo-stage sampling design in which the primary units are geographic regions, the secondary units arepoint-referenced locations, and the measured values are assumed to be a partial realization of a spatialprocess. Traditional geostatistical models do not account for variation attributable to finite populationsampling designs, which can impair inferential performance. On the other hand, design-based estimateswill ignore the spatial dependence in the finite population. This motivates the introduction of geostatisticalprocesses that will enable inference at arbitrary locations in our domain of interest. We demonstrate usingsimulation experiments that process-based finite population sampling models considerably improve modelfit and inference over models that fail to account for spatial correlation. Furthermore, the process basedmodels offer richer inference with spatially interpolated maps over the entire region. We reinforce theseimprovements and demonstrate scaleable inference for groundwater Nitrate levels in the population ofCalifornia Central Valley wells by offering estimates of mean Nitrate levels and their spatially interpolatedmaps.