Lower Bounds for the Integrated and Minimax Risks in Intrinsic Statistical Estimation: A Geometric Approach

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

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Comments for author File: Comments.pdf

Author Response

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Reviewer 2 Report

Comments and Suggestions for Authors

• Clarity and Accessibility:
  While the paper is mathematically rigorous, readability could be improved by briefly restating geometric concepts (e.g., the role of the exponential map, Levi–Civita connection) before their use in proofs.
  Adding a short intuitive explanation after Theorem 2 would help readers grasp the physical or statistical meaning of the derived bounds.

• Notation Consistency:
  In some sections (e.g., 1.2–2.1), the same symbols (e.g., h_{\alpha\beta}, g_{\alpha\beta}​) are used with minimal distinction. It would be helpful to include a concise notation table early in the manuscript.

• Discussion and Interpretation:
  The final section could benefit from a discussion connecting the derived bounds to specific statistical models (e.g., exponential families, Gaussian models). This would demonstrate applicability beyond the Euclidean case.

• Figures or Numerical Examples:
  Including one illustrative example or figure (e.g., comparison of bounds in the Euclidean vs. curved case) would enhance understanding and make the paper more accessible to applied readers.

• English and Style:
  The English is adequate, but some sentences are overly long or literal translations from mathematical formalism. Minor editing for flow and concision would improve readability.

Comments on the Quality of English Language

Minor grammatical adjustments and stylistic refinements—such as simplifying clause structure, ensuring consistent tense usage, and avoiding literal translations of mathematical expressions—would enhance clarity.

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

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Comments for author File: Comments.pdf

Comments on the Quality of English Language

There are many grammatical, spelling, and expression errors in current version, and the writting of the paper should be carefully checked and rewritten for a contextual expression.

Author Response

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Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

Please read the report uploaded.  

Comments for author File: Comments.pdf

Author Response

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I attach a new pdf with most of the changes highligthed with yellow background

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

1. Clarify the novelty of the work

While the manuscript states that it extends previous intrinsic risk results to global risk indices, the introduction would benefit from a more explicit and structured description of:

• What exactly is new compared to Chentsov and the authors’ earlier work [7].

• How the proposed variational approach differs from or generalizes earlier bounds in [2].
  A short paragraph explicitly listing the novel contributions would strengthen the impact.

• 2. Improve the exposition of Theorem 1

  The statement of the intrinsic Cramér–Rao lower bound is correct, but the proof is very dense.
  Consider:

  • Giving more intuition on why the divergence terms appear.

  • Explaining the geometric meaning of the vector fields A and B.

  This will make the result more accessible to readers not fully immersed in information geometry.

• 3. Additional justification in Section 2

  The minimization of the functional Y(B) is central to the paper. The derivation of it could benefit from:

  • A clearer explanation of why the functional is strictly convex.

  • A brief remark on uniqueness of the minimizer.

  • 4. Discussion of curvature conditions

    The results depend on the sign of sectional curvatures, but the discussion is brief. It would be helpful to add:

    • An explanation of the geometric meaning behind the sign of curvature in risk bounds.

    • Examples of statistical models (e.g., exponential families, multivariate normal families) illustrating cases of nonpositive/positive curvature.

    • Appendix clarity:
      The Appendix is useful, but Section 4.1 is long. It may help to split it into subsections (“Manifolds”, “Tensor fields”, “Riemannian metric”, etc.) with clearer headers.

Comments on the Quality of English Language

Minor grammatical adjustments and stylistic refinements—such as simplifying clause structure, ensuring consistent tense usage, and avoiding literal translations of mathematical expressions—would enhance clarity.

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The authors have revised the paper following previous comments and now it satisfies the standard of the journal requirements. Therefore, I suggest to accept the paper at this time.

Author Response

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Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

The marked color is too dark for me to read.  I cannot check the corrections.  Please change the color to be lighter.  Otherwise, I cannot make decision. 

Author Response

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Author Response File: Author Response.pdf

Round 3

Reviewer 2 Report

Comments and Suggestions for Authors

1. Clarify the Conceptual Contribution Early

While the mathematical development is strong, the conceptual novelty could be highlighted more explicitly in the Introduction. In particular, it would help to clearly state:

• What is genuinely new compared to prior intrinsic CR bounds (e.g., [7], Chentsov [2]).

• Why the transition from local intrinsic risk to integrated and minimax intrinsic risks is nontrivial.

• In what sense these new bounds provide insight that cannot be obtained from classical (coordinate-dependent) approaches.

A short paragraph explicitly contrasting local vs. global intrinsic efficiency would improve readability.

2. Motivation for the Variational Formulation

The variational functional

Y(B)=\int_W \Big(\|B\|^2+\tfrac{1}{kn}(\mathrm{div}(B)+a)^2\Big)\,dV is central to the paper. However, the statistical intuition behind this choice could be better explained:

• Why this functional is the “right” object to minimize.

• How it relates to bias-variance trade-offs in intrinsic terms.

• Whether it can be interpreted as a constrained optimization over admissible bias fields.

A short informal explanation before Lemma 1 would be beneficial.

3. Assumptions on Boundary Behavior

Several arguments rely on assuming that the estimator (or its bias field) behaves well on the boundary of W. This is reasonable, but:

• The assumption “bias is negligible on the boundary” should be stated more formally.

• It would help to clarify whether this is a modeling assumption, a technical requirement, or both.

• Are there relevant examples where this assumption fails?

A brief remark discussing the practical implications of boundary conditions would strengthen the paper.

4. Interpretation of Curvature Effects

The dependence of the bounds on sectional curvature is one of the most interesting aspects of the work. However, the discussion remains mostly technical.
Suggestions:

• Add a short interpretative paragraph explaining why negative curvature increases lower bounds and positive curvature relaxes them.

• Relate this behavior to known phenomena in information geometry (e.g., exponential families vs. curved models).

This would make the geometric insight more explicit.

5. Examples Beyond the Gaussian Case

The multivariate normal example is classical and well chosen, but it is also very symmetric.
If feasible, consider:

• Briefly mentioning (even without full computations) how the theory would apply to a non-Euclidean statistical manifold, such as a curved exponential family.

• Alternatively, add a short remark explaining why the Gaussian case already captures the essential phenomena.

This would broaden the perceived applicability of the results.

• Notation consistency

  • The symbols W, B, and S_R​ are heavily used; a small summary table of notation could help readers.

  • Occasionally clarify whether B denotes a region or a bias field (context-dependent).

• Figures

  • Figures 1 and 2 are informative, but captions could explicitly state what improves compared to classical CR bounds.

  • Consider adding a brief sentence in the main text interpreting the asymptotic behavior as R \to \infty.

• References

  • The literature review is solid but somewhat compact. Consider adding 1-2 sentences situating the work relative to recent developments in global efficiency or Bayesian risk bounds on manifolds.

• Language and Style

  • Overall English is good and precise.

  • A light stylistic revision could shorten some long sentences in Sections 3 and 4, improving readability for non-specialists.

Comments on the Quality of English Language

Minor grammatical adjustments and stylistic refinements-such as simplifying clause structure, ensuring consistent tense usage, and avoiding literal translations of mathematical expressions-would enhance clarity.

 

Author Response

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Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

Authors have revised the manuscript and answer all the questions. The manuscript is recommended to be accepted.

Author Response

Thank you very much for your analysis and evaluation