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We present a novel family of nonparametric omnibus tests of the hypothesisthat two unknown but estimable functions are equal in distribution when appliedto the observed data structure. We developed these tests, which represent ageneralization of the maximum mean discrepancy tests described in Gretton etal. [2006], using recent developments from the higher-order pathwisedifferentiability literature. Despite their complex derivation, the associatedtest statistics can be expressed rather simply as U-statistics. We study theasymptotic behavior of the proposed tests under the null hypothesis and underboth fixed and local alternatives. We provide examples to which our tests canbe applied and show that they perform well in a simulation study. As animportant special case, our proposed tests can be used to determine whether anunknown function, such as the conditional average treatment effect, is equal tozero almost surely.

An Omnibus Nonparametric Test of Equality in Distribution for Unknown Functions