  With the widespread application of causal inference, it is increasingly
important to have tools which can test for the presence of causal effects in a
diverse array of circumstances. In this vein we focus on the problem of testing
for \emph{distributional} causal effects, where the treatment affects not just
the mean, but also higher order moments of the distribution, as well as
multidimensional or structured outcomes. We build upon a previously introduced
framework, Counterfactual Mean Embeddings, for representing causal
distributions within Reproducing Kernel Hilbert Spaces (RKHS) by proposing new,
improved, estimators for the distributional embeddings. These improved
estimators are inspired by doubly robust estimators of the causal mean, using a
similar form within the kernel space. We analyse these estimators, proving they
retain the doubly robust property and have improved convergence rates compared
to the original estimators. This leads to new permutation based tests for
distributional causal effects, using the estimators we propose as tests
statistics. We experimentally and theoretically demonstrate the validity of our
tests.
