The Cramér-Rao inequality on singular statistical models I

Abstract

We introduce the notion of the essential tangent bundle of a parametrized measure model and the notion of reduced Fisher metric on a (possibly singular) 2-integrable measure model. Using these notions and a new characterization of $k$-integrable parametrized measure models, we extend the Cramér-Rao inequality to $2$-integrable (possibly singular) statistical models for general $\phi$-estimations, where $\phi$ is a $V$-valued feature function and $V$ is a topological vector space. Thus we derive an intrinsic Cramér-Rao inequality in the most general terms of parametric statistics.

v.2: 28 p, New subsections: 4.4, 4.5, 4.6