Cartan meets Cramér-Rao
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909968192503808 |
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| author | Krishnan, Sunder Ram |
| author_facet | Krishnan, Sunder Ram |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_15612 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cartan meets Cramér-Rao Krishnan, Sunder Ram Statistics Theory Information Theory Signal Processing Differential Geometry 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. |
| title | Cartan meets Cramér-Rao |
| topic | Statistics Theory Information Theory Signal Processing Differential Geometry |
| url | https://arxiv.org/abs/2511.15612 |