Cartan meets Cramér-Rao
Sunder Ram Krishnan
TL;DR
<3-5 sentence high-level summary> This work develops a Cartan-jet geometric foundation for curvature-aware variance bounds in parametric estimation by embedding the square-root model s_θ = √f(·; θ) into a statistical jet bundle and analyzing its prolongations. It proves that algebraic projection conditions underlying CRB and Bhattacharyya-type inequalities are equivalent to m-th order ODE integrability constraints in the jet space, with the vertical Ehresmann component encoding curvature corrections. The authors connect intrinsic holonomy/integrability in the jet tower to extrinsic curvature terms previously derived in Hilbert-space approaches, offering a unified geometric interpretation of higher-order efficiency. Through Gaussian location and natural exponential family examples, the paper illustrates how second-order (and higher) efficiency hinges on the a priori geometry of s_θ and its jets, and how the square-root embedding enables a clean, intrinsic analysis. The framework points to rich future directions, including multi-parameter extensions, holonomy moduli, and ties to intrinsic information geometry.
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 Cramér-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 $s_θ=\sqrt{f(\cdot;θ)}\in L^2(μ)$, we regard $s_θ$ as a section of the statistical bundle $E=Θ\times L^2(μ)$ and study its finite-order prolongations. We point out that the classical algebraic efficiency condition--that the estimator residual $(T-θ)s_θ$ lies in the span of derivatives of $s_θ$ 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 $L^2(μ)$ and intrinsic holonomy properties of statistical sections, yielding a unified geometric interpretation of higher-order information inequalities.