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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.02006 |
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Table of Contents:
- We investigate covariance shrinkage for Hotelling's $T^2$ in the regime where the data dimension $p$ and the sample size $n$ grow in a fixed ratio -- without assuming that the population covariance matrix is spiked or well-conditioned. When $p/n\toϕ\in (0,1)$, we propose a practical finite-sample shrinker that, for any maximum-entropy signal prior and any fixed significance level, (a) asymptotically maximizes power under Gaussian data, and (b) asymptotically saturates the Hanson--Wright lower bound on power in the more general sub-Gaussian case. Our approach is to formulate and solve a variational problem characterizing the optimal limiting shrinker, and to show that our finite-sample method consistently approximates this limit by extending recent local random matrix laws. Empirical studies on simulated and real-world data, including the Crawdad UMich/RSS data set, demonstrate up to a $50\%$ gain in power over leading linear and nonlinear competitors at a significance level of $10^{-4}$.