On the attainment of the Wasserstein--Cramer--Rao lower bound

TL;DR

This paper explores conditions under which estimators achieve the Wasserstein--Cramer--Rao lower bound, focusing on Wasserstein efficiency and proposing a Wasserstein analogue of the maximum likelihood estimator.

Contribution

It establishes conditions for asymptotic Wasserstein efficiency and introduces a Wasserstein estimator that attains this bound in location-scale families.

Findings

Existence of Wasserstein efficient estimators under certain conditions.

Wasserstein estimator is asymptotically efficient in location-scale models.

Connection between Wasserstein efficiency and Wasserstein exponential families.

Abstract

Recently, a Wasserstein analogue of the Cramer--Rao inequality has been developed using the Wasserstein information matrix (Otto metric). This inequality provides a lower bound on the Wasserstein variance of an estimator, which quantifies its robustness against additive noise. In this study, we investigate conditions for an estimator to attain the Wasserstein--Cramer--Rao lower bound (asymptotically), which we call the (asymptotic) Wasserstein efficiency. We show a condition under which Wasserstein efficient estimators exist for one-parameter statistical models. This condition corresponds to a recently proposed Wasserstein analogue of one-parameter exponential families (e-geodesics). We also show that the Wasserstein estimator, a Wasserstein analogue of the maximum likelihood estimator based on the Wasserstein score function, is asymptotically Wasserstein efficient in location-scale families.