Functional ecological inference

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 attempts 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. We then propose the use of 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.