On the attainment of the Wasserstein--Cramer--Rao lower bound
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.
Cite
@article{arxiv.2506.12732,
title = {On the attainment of the Wasserstein--Cramer--Rao lower bound},
author = {Hayato Nishimori and Takeru Matsuda},
journal= {arXiv preprint arXiv:2506.12732},
year = {2025}
}