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Central limit theorem and bootstrap procedure for Wasserstein’s variations with an application to structural relationships between distributions

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  • del Barrio, Eustasio
  • Gordaliza, Paula
  • Lescornel, Hélène
  • Loubes, Jean-Michel

Abstract

Wasserstein barycenters and variance-like criteria based on the Wasserstein distance are used in many problems to analyze the homogeneity of collections of distributions and structural relationships between the observations. We propose the estimation of the quantiles of the empirical process of Wasserstein’s variation using a bootstrap procedure. We then use these results for statistical inference on a distribution registration model for general deformation functions. The tests are based on the variance of the distributions with respect to their Wasserstein’s barycenters for which we prove central limit theorems, including bootstrap versions.

Suggested Citation

  • del Barrio, Eustasio & Gordaliza, Paula & Lescornel, Hélène & Loubes, Jean-Michel, 2019. "Central limit theorem and bootstrap procedure for Wasserstein’s variations with an application to structural relationships between distributions," Journal of Multivariate Analysis, Elsevier, vol. 169(C), pages 341-362.
    • Handle: RePEc:eee:jmvana:v:169:y:2019:i:c:p:341-362
    DOI: 10.1016/j.jmva.2018.09.014
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    References listed on IDEAS

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    1. Freitag, Gudrun & Munk, Axel, 2005. "On Hadamard differentiability in k-sample semiparametric models--with applications to the assessment of structural relationships," Journal of Multivariate Analysis, Elsevier, vol. 94(1), pages 123-158, May.
    2. Agulló-Antolín, Marina & Cuesta-Albertos, J.A. & Lescornel, Hélène & Loubes, Jean-Michel, 2015. "A parametric registration model for warped distributions with Wasserstein’s distance," Journal of Multivariate Analysis, Elsevier, vol. 135(C), pages 117-130.
    3. Axel Munk & Claudia Czado, 1998. "Nonparametric validation of similar distributions and assessment of goodness of fit," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(1), pages 223-241.
    4. Olivier Collier & Arnak S, Dalalyan, 2013. "Curve registration by Nonparametric goodness-of-fit Testing," Working Papers 2013-33, Center for Research in Economics and Statistics.
    5. Max Sommerfeld & Axel Munk, 2018. "Inference for empirical Wasserstein distances on finite spaces," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 80(1), pages 219-238, January.
    6. Alvarez-Esteban, Pedro Cesar & del Barrio, Eustasio & Cuesta-Albertos, Juan Antonio & Matran, Carlos, 2008. "Trimmed Comparison of Distributions," Journal of the American Statistical Association, American Statistical Association, vol. 103, pages 697-704, June.
    7. S. Allassonnière & Y. Amit & A. Trouvé, 2007. "Towards a coherent statistical framework for dense deformable template estimation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(1), pages 3-29, February.
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    1. Vo Nguyen Le Duy & Ichiro Takeuchi, 2023. "Exact statistical inference for the Wasserstein distance by selective inference," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 75(1), pages 127-157, February.

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