Wasserstein Statistics in One-dimensional Location-Scale Model

TL;DR

This paper explores Wasserstein geometry in one-dimensional location-scale models, deriving a new estimator called the W-estimator, which minimizes transportation cost and is shown to be consistent and Fisher efficient in Gaussian cases.

Contribution

It introduces the W-estimator for Wasserstein-based inference in one-dimensional models and analyzes its asymptotic properties, including efficiency.

Findings

W-estimator explicitly minimizes transportation cost.

W-estimator is consistent and asymptotically normal.

Fisher efficiency achieved in Gaussian case.

Abstract

Wasserstein geometry and information geometry are two important structures to be introduced in a manifold of probability distributions. Wasserstein geometry is defined by using the transportation cost between two distributions, so it reflects the metric of the base manifold on which the distributions are defined. Information geometry is defined to be invariant under reversible transformations of the base space. Both have their own merits for applications. In particular, statistical inference is based upon information geometry, where the Fisher metric plays a fundamental role, whereas Wasserstein geometry is useful in computer vision and AI applications. In this study, we analyze statistical inference based on the Wasserstein geometry in the case that the base space is one-dimensional. By using the location-scale model, we further derive the W-estimator that explicitly minimizes the transportation cost from the empirical distribution to a statistical model and study its asymptotic behaviors. We show that the W-estimator is consistent and explicitly give its asymptotic distribution by using the functional delta method. The W-estimator is Fisher efficient in the Gaussian case.