Doubly Robust Kernel Statistics for Testing Distributional Treatment Effects Even Under One Sided Overlap

As causal inference becomes more widespread the importance of having good tools to test for causal effects increases. In this work we focus on the problem of testing for causal effects that manifest in a difference in distribution for treatment and control. We build on work applying kernel methods to causality, considering the previously introduced Counterfactual Mean Embedding framework (\textsc{CfME}). We improve on this by proposing the \emph{Doubly Robust Counterfactual Mean Embedding} (\textsc{DR-CfME}), which has better theoretical properties than its predecessor by leveraging semiparametric theory. This leads us to propose new kernel based test statistics for distributional effects which are based upon doubly robust estimators of treatment effects. We propose two test statistics, one which is a direct improvement on previous work and one which can be applied even when the support of the treatment arm is a subset of that of the control arm. We demonstrate the validity of our methods on simulated and real-world data, as well as giving an application in off-policy evaluation.