Optimal-Transport Based Multivariate Goodness-of-Fit Tests

TL;DR

This paper introduces a new class of multivariate goodness-of-fit tests based on optimal transport theory, which are easy to compute, distribution-free under the null hypothesis, and perform well in finite samples.

Contribution

It develops characteristic-function based tests using optimal measure transport for multivariate ranks, providing a novel, simple, and effective approach for goodness-of-fit testing.

Findings

Tests are distribution-free under the null hypothesis.

Simulation shows competitive performance with existing methods.

Asymptotic theory supports the validity of the tests.

Abstract

Characteristic-function based goodness-of-fit tests are suggested for multivariate observations. The test statistics, which are straightforward to compute, are defined as two-sample criteria measuring discrepancy between multivariate ranks of the original observations and the corresponding ranks obtained from an artificial sample generated from the reference distribution under test. Multivariate ranks are constructed using the theory of the optimal measure transport, thus rendering the tests of a simple null hypothesis distribution-free, while bootstrap approximations are still necessary for testing composite null hypotheses. Asymptotic theory is developed and a simulation study, concentrating on comparisons with previously proposed tests of multivariate normality, demonstrates that the method performs well in finite samples.