An omnibus non‐parametric test of equality in distribution for unknown functions

Alex Luedtke, Marco Carone and Mark J. van der Laan

Journal of the Royal Statistical Society Series B, 2019, vol. 81, issue 1, 75-99

Abstract: We present a novel family of non‐parametric 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 by Gretton and colleagues, using recent developments from the higher order pathwise differentiability literature. Despite their complex derivation, the associated test statistics can be expressed quite simply as U‐statistics. We study the asymptotic behaviour 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.

Date: 2019
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