Functional Ecological Inference

Overview & Synopsis

In this paper we consider the problem of ecological inference when one observes the conditional distributions of Y |W and Z|W from aggregate data and wants to infer the conditional distribution of Y |Z without observing Y and Z in the same sample. First, we show that this problem can be transformed into a linear equation involving operators for which, under suitable regularity assumptions, least squares solutions are available. Then we propose to use the least squares solution with the minimum Hilbert-Schmidt norm, which in our context can be structurally interpreted as the solution with minimum dependence between Y and Z. Interestingly, in the case where the conditioning variable W is discrete and belongs to a finite set, such as the labels of units/groups/cities, the solution of this minimal dependence has a closed form. In the more general case, we use a regularization scheme and show the convergence of our proposed estimator. A numerical evaluation of our procedure is proposed.