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Multivariate Goodness-of-Fit Tests Based on Wasserstein Distance

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  • Marc Hallin
  • Gilles Mordant
  • Johan Segers

Abstract

Goodness-of-fit tests based on the empirical Wasserstein distance are proposed for simple and composite null hypotheses involving general multivariate distributions. This includes the important problem of testing for multivariate normality with unspecified location and covariance and, more generally, testing for elliptical symmetry with given standard radial density, unspecified location and scatter parameters. The calculation of test statistics boils down to solving the well-studied semi-discrete optimal transport problem. Exact critical values can be computed for some important particular cases, such as null hypotheses of ellipticity with given standard radial density and unspecified location and scatter; else, approximate critical values are obtained via parametric bootstrap. Consistency is established, based on a result on the convergence to zero, uniformly over certain families of distributions, of the empirical Wasserstein distance---a novel result of independent interest. A simulation study establishes the practical feasibility and excellent performance of the proposed tests.

Suggested Citation

  • Marc Hallin & Gilles Mordant & Johan Segers, 2020. "Multivariate Goodness-of-Fit Tests Based on Wasserstein Distance," Working Papers ECARES 2020-06, ULB -- Universite Libre de Bruxelles.
  • Handle: RePEc:eca:wpaper:2013/303372
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    Keywords

    Copula; Elliptical distribution; Goodness-of- t; Multivariate normality; Optimal transport; Semi-discrete problem; Skew-t distribution; Wasserstein distance;

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