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With the widespread application of causal inference, it is increasinglyimportant to have tools which can test for the presence of causal effects in adiverse array of circumstances. In this vein we focus on the problem of testingfor \emph{distributional} causal effects, where the treatment affects not justthe mean, but also higher order moments of the distribution, as well asmultidimensional or structured outcomes. We build upon a previously introducedframework, Counterfactual Mean Embeddings, for representing causaldistributions within Reproducing Kernel Hilbert Spaces (RKHS) by proposing new,improved, estimators for the distributional embeddings. These improvedestimators are inspired by doubly robust estimators of the causal mean, using asimilar form within the kernel space. We analyse these estimators, proving theyretain the doubly robust property and have improved convergence rates comparedto the original estimators. This leads to new permutation based tests fordistributional causal effects, using the estimators we propose as testsstatistics. We experimentally and theoretically demonstrate the validity of ourtests.

Doubly Robust Kernel Statistics for Testing Distributional Treatment Effects