Wasserstein-Cramer-Rao Theory of Unbiased Estimation

The quantity of interest in the classical Cramer-Rao theory of unbiased estimation (i.e., the Cramer-Rao lower bound, exact efficiency in 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-Cramer-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 a guarantee that Wasserstein projection estimators achieve 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 new estimators.

This is joint work with Nicolas Garcia Trillos (U Wisconsin) and Adam Jaffe (Columbia) and is based on the paper: https://arxiv.org/pdf/2511.07414.