The Cram\'er-Rao inequality on singular statistical models I

TL;DR

This paper extends the Cramér-Rao inequality to singular and 2-integrable statistical models using new geometric notions like the essential tangent bundle and reduced Fisher metric, providing a general intrinsic bound.

Contribution

It introduces the essential tangent bundle and reduced Fisher metric, enabling the extension of the Cramér-Rao inequality to singular, 2-integrable models for general $\

Findings

Extended Cramér-Rao inequality to singular models

Introduced essential tangent bundle and reduced Fisher metric

Provided an intrinsic, general form of the inequality

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 Cram\'er-Rao inequality to $2$-integrable (possibly singular) statistical models for general $\varphi$-estimations, where $\varphi$ is a $V$-valued feature function and $V$ is a topological vector space. Thus we derive an intrinsic Cram\'er-Rao inequality in the most general terms of parametric statistics.