{"ID":2867135,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.18978","arxiv_id":"2509.18978","title":"Refining Cramér-Rao Bound With Multivariate Parameters: An Extrinsic Geometry Perspective","abstract":"We derive a vector generalization of the curvature-corrected Cramér--Rao bound (CRB) in the nonasymptotic regime using a Hilbert space square-root embedding. Building on previous scalar results, we establish a \\emph{directional} curvature correction derived from the second fundamental form of the model manifold. To obtain matrix-valued refinements, we formulate sufficient conditions for a conservative matrix-level correction using a semidefinite program (SDP) based on sum-of-squares (SOS) relaxations. The framework is rigorously illustrated with two distinct geometries: (i) a curved Gaussian location model, which reveals a characteristic \\textit{pinching effect} where directional bounds vanish along principal axes despite non-zero extrinsic curvature and classical subspace-based bounds using the second-order Bhattacharyya matrix provide overly optimistic variance predictions that fail to account for the manifold's directional topology, and (ii) a spherical multinomial model where the curvature is isotropic. Our results demonstrate that while classical second-order corrections using the Bhattacharyya matrix provide useful benchmarks derived from the local coordinate basis, the proposed directional and SOS-certified bounds offer a more faithful and geometry-consistent representation of the directional sensitivity and fundamental limits of estimation in curved statistical families.","short_abstract":"We derive a vector generalization of the curvature-corrected Cramér--Rao bound (CRB) in the nonasymptotic regime using a Hilbert space square-root embedding. Building on previous scalar results, we establish a \\emph{directional} curvature correction derived from the second fundamental form of the model manifold. To obt...","url_abs":"https://arxiv.org/abs/2509.18978","url_pdf":"https://arxiv.org/pdf/2509.18978v2","authors":"[\"Sunder Ram Krishnan\"]","published":"2025-09-23T13:29:16Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"math.DG\",\"math.PR\"]","methods":"[]","has_code":false}