Wasserstein-Cramér-Rao Theory of Unbiased Estimation

This lecture introduces a new framework for evaluating statistical estimators. While the classical Cramér–Rao theory focuses on the variance of an estimator - that is, how much its estimates vary across different samples - this work instead studies the sensitivity of an estimator. Sensitivity measures how much an estimator changes when the original dataset is subjected to a very small perturbation.

The proposed sensitivity theory is based on Wasserstein geometry and establishes results analogous to those of the classical Cramér–Rao theory. These include a lower bound on the sensitivity of unbiased estimators, a characterization of statistical models in which this bound can be attained exactly, and a proof that Wasserstein projection estimators achieve the bound asymptotically.

Through several statistical examples, the paper demonstrates that this framework reveals new optimality properties of existing estimators and, in some cases, motivates the development of new ones.

Bodhisattva Sen is a Professor of Statistics at Columbia University, New York. His statistical research centers around nonparametrics and large sample theory - nonparametric function estimation (with special emphasis on shape constrained estimation), likelihood and bootstrap based inference in (non-standard) parametric and nonparametric models, optimal transportation and applications in Statistics, etc.