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

arXiv: https://arxiv.org/abs/2506.12732 · HTML: https://mlsanity.com/paper/2506.12732/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 18 references that are in the map.
- First posted on arXiv 15 Jun 2025.

## Builds on (most cited first, 2 of 2)

- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Information Geometry of Wasserstein Statistics on Shapes and Affine Deformations](https://mlsanity.com/paper/2307.12508/) (2023)

## Built on by (most cited first, 1 of 1)

- [Wasserstein-Cram'er-Rao Theory of Unbiased Estimation](https://mlsanity.com/paper/2511.07414/) (2025)

## Abstract (arXiv)

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.