One-dimensional empirical measures, order statistics, and Kantorovich transport distances

By Sergey G. Bobkov, Michel Ledoux. Memoirs of the American Mathematical Society, 2019

314 citations recorded by the source corpus.

This work is devoted to the study of rates of convergence of the empirical measures μn = 1 n ∑n k=1 δXk , n ≥ 1, over a sample (Xk)k≥1 of independent identically distributed real-valued random variables towards the common distribution μ in Kantorovich transport distances Wp. The focus is on finite range bounds on the expected Kantorovich distances E(Wp(μn, μ)) or [ E(W p p (μn, μ)) ]1/p in terms of moments and analytic conditions on the measure μ and its distribution function. The study describes a variety of rates, from the standard one 1 √ n to slower rates, and both lower and upperbounds on E(Wp(μn, μ)) for fixed n in various instances. Order statistics, reduction to uniform samples and analysis of beta distributions, inverse distribution functions, logconcavity are main tools in the investigation. Two detailed appendices collect classical and some new facts on inverse distribution functions and beta distributions and their densities necessary to the investigation.

Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/5d99763e9a0dd9bc6e6101b0de8b000c828327cf

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Citations in the corpus

Multivariate goodness-of-fit tests based on Wasserstein distance (2020, background): “The one-dimensional case is well-studied thanks to the link (1) to empirical quantile processes (del Barrio, Giné and Utzet, 2005; Bobkov and Ledoux, 2019).”
Empirical optimal transport on countable metric spaces: Distributional limits and statistical applications (2017, background): “For a recent survey we refer to Bobkov and Ledoux (2014) and Mason (2016) and references therein.”
On Optimal Matching of Gaussian Samples (2019, background): “This result is achieved in [8] from a characterization of E(W p(μn, μ)) when μ is log-concave in terms of its isoperimetric profile (including further models of interest such as for instance the exponential distribution).”
Generative Modeling Using the Sliced Wasserstein Distance (2018, background): “Using Theorem 4.3 in [5]: where ˆ P � ωf is an independent copy of ˆ P ωf 20), we ﬁrst see how the expectsed Wasserstein distance between two 1-d empirical measures ˆ P ωd and ˆ P ωf can be written in terms of the sets of samples D ω and F ω that they represent (i.e. are induced by).”
Interacting Particle Solutions of Fokker–Planck Equations Through Gradient–Log–Density Estimation (2020, unknown): “X(1) ≤ X(2) ≤ · · · ≤ X(n) and Y(1) ≤ Y(2) ≤ · · · ≤ Y(n) [67].”
Plugin estimation of smooth optimal transport maps (2021, background): “It is well-known that optimal estimation rates under Wasserstein distances differ dramatically in the absence of such a condition (Bobkov and Ledoux, 2019; Weed and Berthet, 2019), and we do not address this setting here.”
Convergence and concentration of empirical measures under Wasserstein distance in unbounded functional spaces (2018, background): “We also extend an argument of [5, 19] from the case of X = R1 to more general cases.”