On parameter estimation with the Wasserstein distance

TL;DR

This paper investigates the asymptotic properties of minimum Wasserstein distance estimators, including in misspecified models, motivated by applications to complex generative models and approximate Bayesian computation.

Contribution

It extends previous results by analyzing the asymptotic behavior of Wasserstein estimators in misspecified settings and discusses practical approximation challenges.

Findings

Establishes asymptotic properties of Wasserstein estimators in misspecified models.

Illustrates estimator behavior through numerical experiments, including ABC applications.

Highlights difficulties in approximating Wasserstein estimators in practice.

Abstract

Statistical inference can be performed by minimizing, over the parameter space, the Wasserstein distance between model distributions and the empirical distribution of the data. We study asymptotic properties of such minimum Wasserstein distance estimators, complementing results derived by Bassetti, Bodini and Regazzini in 2006. In particular, our results cover the misspecified setting, in which the data-generating process is not assumed to be part of the family of distributions described by the model. Our results are motivated by recent applications of minimum Wasserstein estimators to complex generative models. We discuss some difficulties arising in the approximation of these estimators and illustrate their behavior in several numerical experiments. Two of our examples are taken from the literature on approximate Bayesian computation and have likelihood functions that are not analytically tractable. Two other examples involve misspecified models.