An Omnibus Nonparametric Test of Equality in Distribution for Unknown Functions

TL;DR

This paper introduces a new family of nonparametric tests for comparing unknown functions in distribution, extending MMD tests with higher-order differentiability, and demonstrates their effectiveness through theoretical analysis and simulations.

Contribution

It develops a generalized omnibus test for equality in distribution of unknown functions, utilizing higher-order pathwise differentiability and U-statistics, applicable to various data structures.

Findings

Tests perform well in simulations

Can determine if an unknown function equals zero almost surely

Theoretical properties under null and alternative hypotheses

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.