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

TL;DR

This paper develops a bootstrap-based method for estimating quantiles of Wasserstein variation, enabling statistical inference on distribution models, supported by central limit theorems including bootstrap versions.

Contribution

It introduces a bootstrap procedure for Wasserstein variation quantiles and proves central limit theorems for these measures, enhancing statistical inference capabilities.

Findings

Bootstrap method accurately estimates Wasserstein variation quantiles.

Central limit theorems established for Wasserstein barycenter variance.

Method applied to distribution registration models with deformation functions.

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.