Dr. Nicolás Garcia Trillos from the University of Wisconsin–Madison

Title: Wasserstein-Cramér-Rao Theory of unbiased estimation and tradeoffs between accuracy and robustness of estimators.

Abstract: In this talk, we will be interested in a quantity that 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. I’ll present a collection of results which are analogous to the classical case: a Wasserstein-Cramér-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. For both the classical and Wasserstein settings a strong geometric intuition guides the corresponding statistical theories. This same geometric perspective allow us to formulate and answer a natural and important question: how can we construct estimators that, at least asymptotically, balance between accuracy (variance) and robustness (sensitivity) optimally?