# Wasserstein-Cram'er-Rao Theory of Unbiased Estimation

arXiv: https://arxiv.org/abs/2511.07414 · HTML: https://mlsanity.com/paper/2511.07414/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 72 references that are in the map.
- First posted on arXiv 10 Nov 2025.

## Builds on (most cited first, 16 of 16)

- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [Distributionally Robust Stochastic Optimization with Wasserstein Distance](https://mlsanity.com/paper/1604.02199/) (2016)
- [Stability](https://mlsanity.com/paper/1310.0150/) (2013)
- [Veridical Data Science](https://mlsanity.com/paper/1901.08152/) (2019)
- [Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections](https://mlsanity.com/paper/2107.01718/) (2021)
- [Distributionally Robust Optimization and Robust Statistics](https://mlsanity.com/paper/2401.14655/) (2024)
- [When Optimal Transport Meets Information Geometry](https://mlsanity.com/paper/2206.14791/) (2022)
- [Stability Bounds for Smooth Optimal Transport Maps and their Statistical Implications](https://mlsanity.com/paper/2502.12326/) (2025)
- [On the regularized risk of distributionally robust learning over deep neural networks](https://mlsanity.com/paper/2109.06294/) (2021)
- [Estimation Stability with Cross Validation (ESCV)](https://mlsanity.com/paper/1303.3128/) (2013)
- [On the Uniqueness of Kantorovich Potentials](https://mlsanity.com/paper/2201.08316/) (2022)
- [Hellinger-Kantorovich Gradient Flows: Global Exponential Decay of Entropy Functionals](https://mlsanity.com/paper/2501.17049/) (2025)
- [Quantile-constrained Wasserstein projections for robust interpretability of numerical and machine learning models](https://mlsanity.com/paper/2209.11539/) (2022)
- [Information Geometry of Wasserstein Statistics on Shapes and Affine Deformations](https://mlsanity.com/paper/2307.12508/) (2023)
- [Asymptotic analysis of the jittering kernel density estimator](https://mlsanity.com/paper/1705.05431/) (2017)
- [On the attainment of the Wasserstein--Cramer--Rao lower bound](https://mlsanity.com/paper/2506.12732/) (2025)

## Built on by (most cited first, 2 of 2)

- [Unfolding with a Wasserstein Loss](https://mlsanity.com/paper/2603.20903/) (2026)
- [Robust Statistical Estimators with Bounded Empirical Sensitivity](https://mlsanity.com/paper/2605.21860/) (2026)

## Abstract (arXiv)

The quantity of interest in the classical Cram'er-Rao theory of unbiased estimation (e.g., the Cram'er-Rao lower bound, its exact attainment for exponential families, and asymptotic efficiency of maximum likelihood estimation) is the variance, which represents the instability of an estimator when its value is compared to the value for an independently-sampled data set from the same distribution. In this paper we are interested in a quantity which represents the instability of an estimator when its value is compared to the value for an infinitesimal additive perturbation of the original data set; we refer to this as the "sensitivity" of an estimator. The resulting theory of sensitivity is based on the Wasserstein geometry in the same way that the classical theory of variance is based on the Fisher-Rao (equivalently, Hellinger) geometry, and this insight allows us to determine a collection of results which are analogous to the classical case: a Wasserstein-Cram'er-Rao lower bound for the sensitivity of any unbiased estimator, a characterization of models in which there exist unbiased estimators achieving the lower bound exactly, and some concrete results that show that the Wasserstein projection estimator achieves the lower bound asymptotically. We use these results to treat many statistical examples, sometimes revealing new optimality properties for existing estimators and other times revealing entirely new estimators.