Title: 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
$\varphi$-estimations, where $\varphi$ 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.