  We present a novel family of nonparametric omnibus tests of the hypothesis
that two unknown but estimable functions are equal in distribution when applied
to the observed data structure. We developed these tests, which represent a
generalization of the maximum mean discrepancy tests described in Gretton et
al. [2006], using recent developments from the higher-order pathwise
differentiability literature. Despite their complex derivation, the associated
test statistics can be expressed rather simply as U-statistics. We study the
asymptotic behavior of the proposed tests under the null hypothesis and under
both fixed and local alternatives. We provide examples to which our tests can
be applied and show that they perform well in a simulation study. As an
important special case, our proposed tests can be used to determine whether an
unknown function, such as the conditional average treatment effect, is equal to
zero almost surely.
