# An Omnibus Nonparametric Test of Equality in Distribution for Unknown Functions 

Alexander R. Luedtke $^{1,2}$, Mark J. van der Laan$^{3}$, and Marco Carone4,1 

1 Vaccine and Infectious Disease Division, Fred Hutchinson Cancer Research Center, USA 

$^{2}$Public Health Sciences Division, Fred Hutchinson Cancer Research Center, USA 

$^{3}$Division of Biostatistics, University of California, Berkeley, USA 

$^{4}$Department of Biostatistics, University of Washington, USA 

## Abstract 

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. 

Kewyords: higher order pathwise differentiability, maximum mean discrepancy, omnibus test, equality in distribution, infinite dimensional parameter. 

