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Refining Cramér-Rao Bound With Multivariate Parameters: An Extrinsic Geometry Perspective

Sunder Ram Krishnan

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.

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.

Paper Structure

This paper contains 13 sections, 8 theorems, 113 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Setup, notation, and assumptions
  3. Projection proof of the classical matrix CRB
  4. Induced connection and the second fundamental form
  5. Directional curvature-corrected CRB
  6. Matrix curvature correction
  7. On "directional" vs "matrix" curvature corrections
  8. A conservative matrix correction
  9. Higher-order jet space refinements
  10. Two detailed examples
  11. Numerical evaluation and the geometry of the curvature bound
  12. Synthesis of results: Degenerate vs. Non-degenerate geometry
  13. Conclusion and future work

Key Result

Lemma 1

The $m$-th order embedded derivative vectors $\eta_\alpha$ can be expressed as a product of the square-root density $s$ and a polynomial in the lower-order score functions $Y_\beta$. Specifically, for orders $1$ and $2$: Generally, for any multi-index $\alpha$, $\eta_\alpha = s \cdot \mathcal{P}_\alpha(Y_{\beta \le \alpha})$, where $\mathcal{P}_\alpha$ is the polynomial determined by the Faà di B

Figures (2)

  • Figure 1: The "pinching effect" in the Gaussian location model. The directional bound $\mathcal{R}(v)$ (black solid) vanishes along the coordinate axes, revealing the non-quadratic nature of the curvature correction. The classical analytical $\Delta_\mathrm{B}$ matrix (red dashed) violates the directional bound in these regions, demonstrating that a global matrix correction can be overly optimistic. This necessitates the use of our SOS-SDP approach to find a truly conservative, certified matrix $\Delta$.
  • Figure 2: Comparison of directional and matrix bounds across differing manifold geometries. In the Gaussian case (a), the analytical $\Delta_\mathrm{B}$ violates the directional bound $\mathcal{R}(v)$ at the axes, leading to $\Delta=0$ in the (SOS-SDP). In the spherical case (b), all bounds coincide.

Theorems & Definitions (29)

  • Lemma 1: Faà di Bruno Expansion and Statistical Scores
  • proof
  • Remark 1: Statistical Significance of the Expansion
  • Proposition 3: Projection inequality
  • proof
  • Proposition 4: Identification of the projection matrix: $B=J^{-1}$
  • proof
  • Definition 1: Induced connection
  • Definition 2: Second fundamental form
  • Proposition 5: Symmetry of $\Pi$
  • ...and 19 more