Downscaling direct estimators in small area models

Statistical modeling often faces the problem of missing information. This could be due either to data privacy reasons or to the lack of data detection in a specific territorial domain. In this paper, we show a methodology to downscale estimators from larger to smaller territorial domains. Our method is embedded in a small area estimation context, in particular at area-level scale. The proposed method allows us to exploit information from the direct estimators measured in the larger territorial domains that are planned by the sampling design combined with some auxiliary variables at a finer unplanned territorial level. The resulting estimators are a convex linear combination on a different scale of the direct estimators and the synthetic parts based on a statistical model. As an illustration, we use this methodology to highlight differences in the school performances of Italian students. We downscale validated estimators from the planned regional to the unplanned provincial territorial level with the scope to provide more reliable estimators in such a spatial domain than the existing unreliable data. We use data from the Italian National Institute for the Evaluation of the School System (INVALSI), which releases validated sample data at the regional spatial scale, and not validated census data at the provincial level. We use the latter as auxiliary information to predict the provincial scores.