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Graphical and uniform consistency of estimated optimal transport plans

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  • Segers, Johan

    (Université catholique de Louvain, LIDAM/ISBA, Belgium)

Abstract

A general theory is provided delivering convergence of maximal cyclically monotone mappings containing the supports of coupling measures of sequences of pairs of possibly random probability measures on Euclidean space. The theory is based on the identification of such a mapping with a closed subset of a Cartesian product of Euclidean spaces and leveraging tools from random set theory. Weak convergence in the appropriate Fell space together with the maximal cyclical monotonicity then automatically yields local uniform convergence of the associated mappings. Viewing such mappings as optimal transport plans between probability measures with respect to the squared Euclidean distance as cost function yields consistency results for notions of multivariate ranks and quantiles based on optimal transport, notably the empirical center-outward distribution and quantile functions.

Suggested Citation

  • Segers, Johan, 2022. "Graphical and uniform consistency of estimated optimal transport plans," LIDAM Discussion Papers ISBA 2022022, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    • Handle: RePEc:aiz:louvad:2022022
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    References listed on IDEAS

    as
    1. Hongjian Shi & Mathias Drton & Fang Han, 2022. "Distribution-Free Consistent Independence Tests via Center-Outward Ranks and Signs," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 117(537), pages 395-410, January.
    2. Bucher, Axel & Segers, Johan & Volgushev, Stanislav, 2014. "When uniform weak convergence fails: empirical processes for dependence functions via epi- and hypographs," LIDAM Reprints ISBA 2014018, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    3. Eustasio del Barrio & Alberto González-Sanz & Marc Hallin, 2022. "Nonparametric Multiple-Output Center-Outward Quantile Regression," Working Papers ECARES 2022-10, ULB -- Universite Libre de Bruxelles.
    4. Marc Hallin & Daniel Hlubinka & Šárka Hudecová, 2023. "Efficient Fully Distribution-Free Center-Outward Rank Tests for Multiple-Output Regression and MANOVA," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 118(543), pages 1923-1939, July.
    5. Beirlant, J. & Buitendag, S. & del Barrio, E. & Hallin, M. & Kamper, F., 2020. "Center-outward quantiles and the measurement of multivariate risk," Insurance: Mathematics and Economics, Elsevier, vol. 95(C), pages 79-100.
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