Inference for empirical Wasserstein distances on finite spaces

By Max Sommerfeld, Axel Munk. 2016

185 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/707328d843ea38397020022e0e39c94c75e27043

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In Applied Mathematics

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Citations in the corpus

Fréchet means and Procrustes analysis in Wasserstein space (2017, background): “The Procrustes maps can also be used in order to produce a Principal Component Analysis, capturing the main modes of deformation-based variation (Bigot et al. [13], Panaretos & Zemel [61]; Huckemann, Munk & Hotz [42], Wang et al. [75]).”
Empirical optimal transport on countable metric spaces: Distributional limits and statistical applications (2017, background): “5 in Sommerfeld and Munk (2016) for finite spaces to countable spaces X .”
Plugin estimation of smooth optimal transport maps (2021, background): “Key exceptions are obtained when P and Q are one-dimensional (Munk and Czado, 1998; Freitag and Munk, 2005) or countable (Sommerfeld and Munk, 2018; Tameling et al., 2019), in which case non-degenerate limiting distributions for the process W2(Pn, Qm) − W2(P,Q) are known up to suitable scaling.”
Multivariate goodness-of-fit tests based on Wasserstein distance (2020, methodology): “Also for discrete distributions, non-degenerate limit distributions are known (Sommerfeld and Munk, 2018; Tameling, Sommerfeld and Munk, 2019).”
Convergence and concentration of empirical measures under Wasserstein distance in unbounded functional spaces (2018, background): “See [33] for a more thorough historical review of Wasserstein distances and their applications in statistics, and [40] for further details about Wasserstein distances and optimal transport in a broader context.”
Central limit theorems for empirical transportation cost in general dimension (2017, methodology): “To our best knowledge the only work that deals with the issue of distributional limit laws for the transportation cost between empirical measures is [18] (see Theorem 1 there).”
Statistical Aspects of Wasserstein Distances (2018, methodology): “Sommerfeld & Munk (2018) highlight the implications of the non-standard delta method for the bootstrap, whose consistency requires subsampling.”