Doubly Flexible Estimation under Label Shift

In studies ranging from clinical medicine to policy research, complete data are usually available from a population mathscrP, but the quantity of interest is often sought for a related but different population mathscrQ which only has partial data. In this paper, we consider the setting that both outcome Y and covariate bf X are available from mathscrP whereas only bf X is available from mathscrQ, under the so-called label shift assumption, i.e., the conditional distribution of bf X given Y remains the same across the two populations. To estimate the parameter of interest in mathscrQ via leveraging the information from mathscrP, the following three ingredients are essential: (a) the common conditional distribution of bf X given Y, (b) the regression model of Y given bf X in mathscrP, and (c) the density ratio of Y between the two populations. We propose an estimation procedure that only needs standard nonparametric technique to approximate the conditional expectations with respect to (a), while by no means needs an estimate or model for (b) or (c); i.e., doubly flexible to the possible model misspecifications of both (b) and (c). This is conceptually different from the well-known doubly robust estimation in that, double robustness allows at most one model to be misspecified whereas our proposal can allow both (b) and (c) to be misspecified. This is of particular interest in our setting because estimating (c) is difficult, if not impossible, by virtue of the absence of the Y-data in mathscrQ. Furthermore, even though the estimation of (b) is sometimes off-the-shelf, it can face curse of dimensionality or computational challenges. We develop the large sample theory for the proposed estimator, and examine its finite-sample performance through simulation studies as well as an application to the MIMIC-III database.