The Cramér-rao inequality on singular statistical models

Abstract

We introduce the notions of essential tangent space and reduced Fisher metric and extend the classical Cramér-Rao inequality to 2-integrable (possibly singular) statistical models for general ϕ-estimators, where ϕ is a V-valued feature function and V is a topological vector space. We show the existence of a ϕ-efficient estimator on strictly singular statistical models associated with a finite sample space and on a class of infinite dimensional exponential models that have been discovered by Fukumizu. We conclude that our general Cramér-Rao inequality is optimal.