Central limit theorem and bootstrap procedure for Wasserstein’s variations with an application to structural relationships between distributions

Abstract

Wasserstein barycenters and variance-like criteria based on the Wasserstein distance are used in many problems to analyze the homogeneity of collections of distributions and structural relationships between the observations. We propose the estimation of the quantiles of the empirical process of Wasserstein's variation using a bootstrap procedure. We then use these results for statistical inference on a distribution registration model for general deformation functions.The tests are based on the variance of the distributions with respect to their Wasserstein's barycenters for which we prove central limit theorems, including bootstrap versions.In cases where G is a parametric class, estimation of the warping functions is studied in [2]. However, estimation/registration procedures may lead to inconsistent conclusions if the chosen deformation class G is too small. It is, therefore, important to be able to assess the fit to the deformation model given by a particular choice of G. This is the main goal of this paper. We note that within this framework, statistical inference on deformation models for distributions has been studied first in [20]. Here we provide a different approach which allows to deal with more general deformation classes.The pioneering works [15,25] study the existence of relationships between distributions F and G by using a discrepancy measure ∆(F, G) between them which is built using the Wasserstein distance. The authors consider the assumption H 0 : ∆(F, G) > ∆ 0 versus H a : ∆(F, G) ≤ ∆ 0 for a chosen threshold ∆ 0 . Thus when the null hypothesis is rejected, there is statistical evidence that the two distributions are similar with respect to the chosen criterion. In this same vein, we define a notion of variation of distributions using the Wasserstein distance, W r , in the set W r (R d ) of probability measures with finite rth moments, where r ≥ 1. This notion generalizes the concept of variance for random distributions over R d . This quantity can be defined aswhich measures the spread of the distributions. Then, to measure closeness to a deformation model, we take a look at the minimal variation among warped distributions, a quantity that we could consider as a minimal alignment cost. Under some mild conditions, a deformation model holds if and only if this minimal alignment cost is null and we can base our assessment of a deformation model on this quantity. As in [15,25], we provide results (a Central Limit Theorem and bootstrap versions) that enable to reject that the minimal alignment cost exceeds some threshold, and hence to conclude that it is below that threshold. Our results are given in a setup of general, nonparametric classes of warping functions. We also provide results in the somewhat more restrictive setup where one is interested in the more classical goodness-of-fit problem for the deformation model. Note that a general Central Limit Theorem is available for the Wasserstein distance in [18].The paper is organized as follows. The main facts about Wasserstein variation are present...

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