On Higher-Order Geometric Refinements of Classical Covariance Asymptotics: An Approach via Intrinsic and Extrinsic Information Geometry

Classical Fisher-information asymptotics describe the covariance of regular efficient estimators through the local quadratic approximation of the log-likelihood, and thus capture first-order geometry only. In curved models, including mixtures, curved exponential families, latent-variable models, and manifold-constrained parameter spaces, finite-sample behavior can deviate systematically from these predictions. We develop a coordinate-invariant, curvature-aware refinement by viewing a regular parametric family as a Riemannian manifold \((Θ,g)\) with Fisher--Rao metric, immersed in \(L^2(μ)\) through the square-root density map. Under suitable regularity and moment assumptions, we derive an...

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