Inference for empirical Wasserstein distances on finite spaces

Journal of the Royal Statistical Society: Series B (Statistical Methodology) · 2018 · Volume 80 · Issue 1

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.

Keywords

Bootstrap, Central limit theorem, Directional Hadamard derivative, Hypothesis testing, Optimal transport, Wasserstein distance

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