2025/04/14 by William L. Cook, Cook, William
Computer Science · Mathematics · #53B20 #62F12 #90C25 #Advanced Statistical Methods and Models #Data Analysis with R #FOS: Computer and information sciences #G.1.6 #G.3 #Other Computer Science (cs.OH) #Statistics Education and Methodologies
paper · pdf · doi:10.48550/arxiv.2504.10667
openalex publication_date 2025/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Optimal statistical decisions should transcend the language used to describe them. Yet, how do we guarantee that the choice of coordinates - the parameterisation of an optimisation problem - does not subtly dictate the solution? This paper reveals a fundamental geometric invariance principle. We first analyse the optimal combination of two asymptotically normal estimators under a strictly convex trace-AMSE risk. While methods for finding optimal weights are known, we prove that the resulting optimal estimator is invariant under direct affine reparameterisations of the weighting scheme. This exemplifies a broader principle we term meta-equivariance: the unique minimiser of any strictly convex, differentiable scalar objective over a matrix space transforms covariantly under any invertible affine reparameterisation of that space. Distinct from classical statistical equivariance tied to data symmetries, meta-equivariance arises from the immutable geometry of convex optimisation itself. It guarantees that optimality, in these settings, is not an artefact of representation but an intrinsic, coordinate-free truth.