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Inference for empirical Wasserstein distances on finite spaces

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  • Max Sommerfeld
  • Axel Munk

Abstract

The Wasserstein distance is an attractive tool for data analysis but statistical inference is hindered by the lack of distributional limits. To overcome this obstacle, for probability measures supported on finitely many points, we derive the asymptotic distribution of empirical Wasserstein distances as the optimal value of a linear programme with random objective function. This facilitates statistical inference (e.g. confidence intervals for sample‐based Wasserstein distances) in large generality. Our proof is based on directional Hadamard differentiability. Failure of the classical bootstrap and alternatives are discussed. The utility of the distributional results is illustrated on two data sets.

Suggested Citation

  • 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.
    • Handle: RePEc:bla:jorssb:v:80:y:2018:i:1:p:219-238
    DOI: 10.1111/rssb.12236
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    Cited by:

    1. Marcel Klatt & Axel Munk & Yoav Zemel, 2022. "Limit laws for empirical optimal solutions in random linear programs," Annals of Operations Research, Springer, vol. 315(1), pages 251-278, August.
    2. Valentin Hartmann & Dominic Schuhmacher, 2020. "Semi-discrete optimal transport: a solution procedure for the unsquared Euclidean distance case," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(1), pages 133-163, August.
    3. Espen Bernton & Pierre E. Jacob & Mathieu Gerber & Christian P. Robert, 2019. "Approximate Bayesian computation with the Wasserstein distance," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 81(2), pages 235-269, April.
    4. Tengyuan Liang, 2020. "Estimating Certain Integral Probability Metrics (IPMs) Is as Hard as Estimating under the IPMs," Working Papers 2020-153, Becker Friedman Institute for Research In Economics.
    5. Mario Ghossoub & Jesse Hall & David Saunders, 2020. "Maximum Spectral Measures of Risk with given Risk Factor Marginal Distributions," Papers 2010.14673, arXiv.org.
    6. 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.
    7. Lin Lin & Wei Shi & Jianbo Ye & Jia Li, 2023. "Multisource single‐cell data integration by MAW barycenter for Gaussian mixture models," Biometrics, The International Biometric Society, vol. 79(2), pages 866-877, June.

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