Minimax Linear Estimation of the Retargeted Mean

Evaluating treatments received by one population for application to a different target population of scientific interest is a central problem in causal inference from observational studies. We study the minimax linear estimator of the treatment-specific mean outcome on a target population and provide a theoretical basis for inference based on it. In particular, we provide a justification for the common practice of ignoring bias when building confidence intervals with these linear estimators. Focusing on the case that the class of the unknown outcome function is the unit ball of a reproducing kernel Hilbert space, we show that the resulting linear estimator is asymptotically optimal under conditions only marginally stronger than those used with augmented estimators. We establish bounds attesting to the estimator's good finite sample properties. In an extensive simulation study, we observe promising performance of the estimator throughout a wide range of sample sizes, noise levels, and levels of overlap between the covariate distributions of the treated and target populations.