An Omnibus Nonparametric Test of Equality in Distribution for Unknown Functions
This provides a nonparametric test for equality in distribution of functions, useful in fields like causal inference for assessing conditional average treatment effects, but it is incremental as it builds on prior work.
The authors tackled the problem of testing whether two unknown functions are equal in distribution, generalizing existing maximum mean discrepancy tests using higher-order pathwise differentiability, and demonstrated good performance in simulations.
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.