Local Optimality and Rigidity of Frobenius Tests for Dense High-Dimensional Covariance Alternatives
Abstract
We study identity testing for high-dimensional covariance matrices against dense alternatives of unknown direction, with p/n → γ. Along a globally positive quadratic precision path, mixing Gaussian alternatives over a Gaussian Orthogonal Ensemble direction yields a contiguous experiment whose log likelihood reduces to the corrected Frobenius statistic; its upper-tail test attains the limiting weighted-power envelope at every fixed strength. Fixing the prior's Frobenius radius perturbs the mixture by only O(p−1/2) in total variation, and exact whitening carries the experiment, the statistic, and its null law to any known null covariance. Separately, under a product-coordinate null, feasibility needs only 4+η moments, plus identical distributions over time when means are estimated; studentization and an exact degrees-of-freedom correction preserve the local power. A stability inequality turns near-envelope attainment into null agreement with the Frobenius rule, so uniform noninferiority on the typical dense bulk precludes gains at any contiguous alternative. For trace-matched rank-one alternatives, the corrected statistic is the first likelihood direction when ϑn → 0 and nϑn → ∞; at fixed strength, the log likelihood ratio in the Onatski-Moreira-Hallin fixed-spike benchmark is governed by a richer linear spectral statistic below the Baik-Ben Arous-Peche threshold, while eigenvalue separation permits cost-free largest-eigenvalue enhancement above it. Simulations illustrate the theory.
Related
- On Frobenius rigidity for motivic cohomology
- BCC optimality for the three-dimensional Coulomb lattice energy
- Sharp Harmonic-Mean Bounds for Neumann Eigenvalues on Riemannian Surfaces and Counterexamples to Geodesic Ball Optimality on Higher-Dimensional Spheres
- Exact Bayes Regret and Asymptotic Optimality in High-Dimensional Gaussian Bandits
- High-Dimensional Regularization of the Spatial Sign Covariance Matrix for Robust Shape Estimation
BibTeX
@misc{hansen2026local,
title = {{Local Optimality and Rigidity of Frobenius Tests for Dense High-Dimensional Covariance Alternatives}},
author = {Peter Reinhard Hansen and Werner Ploberger and Chen Tong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27200},
eprint = {2609.27200},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27200},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Local Optimality and Rigidity of Frobenius Tests for Dense High-Dimensional Covariance Alternatives
AU - Peter Reinhard Hansen
AU - Werner Ploberger
AU - Chen Tong
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27200
UR - https://arxiv.org/abs/2609.27200
AB - We study identity testing for high-dimensional covariance matrices against dense alternatives of unknown direction, with p/n → γ. Along a globally positive quadratic precision path, mixing Gaussian alternatives over a Gaussian Orthogonal Ensemble direction yields a contiguous experiment whose log likelihood reduces to the corrected Frobenius statistic; its upper-tail test attains the limiting weighted-power envelope at every fixed strength. Fixing the prior's Frobenius radius perturbs the mixture by only O(p^−1/2) in total variation, and exact whitening carries the experiment, the statistic, and its null law to any known null covariance. Separately, under a product-coordinate null, feasibility needs only 4+η moments, plus identical distributions over time when means are estimated; studentization and an exact degrees-of-freedom correction preserve the local power. A stability inequality turns near-envelope attainment into null agreement with the Frobenius rule, so uniform noninferiority on the typical dense bulk precludes gains at any contiguous alternative. For trace-matched rank-one alternatives, the corrected statistic is the first likelihood direction when ϑ_n → 0 and nϑ_n → ∞; at fixed strength, the log likelihood ratio in the Onatski-Moreira-Hallin fixed-spike benchmark is governed by a richer linear spectral statistic below the Baik-Ben Arous-Peche threshold, while eigenvalue separation permits cost-free largest-eigenvalue enhancement above it. Simulations illustrate the theory.
ER -