Cartan meets Cram\'er-Rao

Abstract

A Cartan-geometric, jet bundle formulation of curvature-aware variance bounds in parametric statistical estimation is developed. Building on our earlier extrinsic Hilbert space approach to the Cram\'er-Rao and Bhattacharyya-type inequalities, we show that the curvature corrections induced by the square root embedding of a statistical model admit a canonical intrinsic interpretation via jet geometry and Cartan's prolongation theory. For a scalar-parameter family with square root map $s_\theta=\sqrt{f(\cdot;\theta)}\in L^2(\mu)$, we regard $s_\theta$ as a section of the statistical bundle $E=\Theta\times L^2(\mu)$ and study its finite-order prolongations. We point out that the classical algebraic efficiency condition--that the estimator residual $(T-\theta)s_\theta$ lies in the span of derivatives of $s_\theta$ up to order $m$--is equivalent to the existence of a linear ordinary differential equation (ODE) of order $m$ satisfied by the square root map. Geometrically, this means the prolonged section lies in an ODE-defined submanifold of the jet bundle and is an integral curve of the restricted Cartan vector field. The obstruction to such finite-order integrability is identified with the vertical component of the canonical Ehresmann connection on the jet tower, which coincides with the curvature correction term in variance bounds. This establishes a direct correspondence between algebraic projection conditions in $L^2(\mu)$ and intrinsic holonomy properties of statistical sections, yielding a unified geometric interpretation of higher-order information inequalities.