Open Access
2021 Multivariate goodness-of-fit tests based on Wasserstein distance
Marc Hallin, Gilles Mordant, Johan Segers
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Electron. J. Statist. 15(1): 1328-1371 (2021). DOI: 10.1214/21-EJS1816


Goodness-of-fit tests based on the empirical Wasserstein distance are proposed for simple and composite null hypotheses involving general multivariate distributions. For group families, the procedure is to be implemented after preliminary reduction of the data via invariance. This property allows for calculation of exact critical values and p-values at finite sample sizes. Applications include testing for location–scale families and testing for families arising from affine transformations, such as elliptical distributions with given standard radial density and unspecified location vector and scatter matrix. A novel test for multivariate normality with unspecified mean vector and covariance matrix arises as a special case. For more general parametric families, we propose a parametric bootstrap procedure to calculate critical values. The lack of asymptotic distribution theory for the empirical Wasserstein distance means that the validity of the parametric bootstrap under the null hypothesis remains a conjecture. Nevertheless, we show that the test is consistent against fixed alternatives. To this end, we prove a uniform law of large numbers for the empirical distribution in Wasserstein distance, where the uniformity is over any class of underlying distributions satisfying a uniform integrability condition but no additional moment assumptions. The calculation of test statistics boils down to solving the well-studied semi-discrete optimal transport problem. Extensive numerical experiments demonstrate the practical feasibility and the excellent performance of the proposed tests for the Wasserstein distance of order p=1 and p=2 and for dimensions at least up to d=5. The simulations also lend support to the conjecture of the asymptotic validity of the parametric bootstrap.

Funding Statement

J. Segers gratefully acknowledges funding from FNRS-F.R.S. grant CDR J.0146.19.


The authors gratefully acknowledge the remarks and comments made by the reviewers that greatly helped improve the paper.


Download Citation

Marc Hallin. Gilles Mordant. Johan Segers. "Multivariate goodness-of-fit tests based on Wasserstein distance." Electron. J. Statist. 15 (1) 1328 - 1371, 2021.


Received: 1 October 2020; Published: 2021
First available in Project Euclid: 16 March 2021

Digital Object Identifier: 10.1214/21-EJS1816

Keywords: copula , elliptical distribution , Goodness-of-fit , group families , multivariate normality , Optimal transport , semi-discrete problem , skew-t distribution , Wasserstein distance

Vol.15 • No. 1 • 2021
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