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

TL;DR

This paper develops a Wasserstein-based sensitivity theory for unbiased estimators, analogous to classical variance-based Cramér-Rao theory, providing bounds, characterizations, and new insights into estimator optimality.

Contribution

It introduces a Wasserstein-Cramér-Rao framework for estimator sensitivity, extending classical theory to a new geometric setting and identifying conditions for optimal unbiased estimators.

Findings

Derives a lower bound for estimator sensitivity in Wasserstein geometry.

Characterizes models with exactly unbiased estimators achieving the bound.

Shows the Wasserstein projection estimator attains the bound asymptotically.

Abstract

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.