AI 中文总结
本文研究高维协方差矩阵的恒等性检验，针对密集备择，提出修正的Frobenius统计量并证明其达到加权功效包络，同时揭示其局部最优性与刚性条件。
AI 中文摘要
我们研究高维协方差矩阵的恒等性检验，针对方向未知的密集备择假设，其中 $p/n \to γ$。沿着一条全局正定的二次精度路径，将高斯备择假设在高斯正交系综方向上混合，产生一个邻接实验，其对数似然简化为修正的Frobenius统计量；其右尾检验在任意固定强度下达到极限加权功效包络。固定先验的Frobenius半径仅以 $O(p^{-1/2})$ 的总变差扰动混合，且精确白化将实验、统计量及其零分布转移到任意已知的零协方差。另外，在乘积坐标零假设下，可行性仅需 $4+η$ 阶矩，当估计均值时还需时间上的同分布；学生化和精确自由度校正保持局部功效。一个稳定性不等式将近似包络达到转化为与Frobenius规则的零一致性，因此在典型密集块上的均匀非劣性排除了任何邻接备择下的增益。对于迹匹配的秩一备择，当 $\vartheta_n \to 0$ 且 $n\vartheta_n \to \infty$ 时，修正统计量是第一个似然方向；在固定强度下，Onatski-Moreira-Hallin固定尖峰基准中的对数似然比在Baik-Ben Arous-Peche阈值之下由更丰富的线性谱统计量控制，而特征值分离允许在阈值之上进行无成本的最大特征值增强。模拟验证了理论。
英文摘要
We study identity testing for high-dimensional covariance matrices against dense alternatives of unknown direction, with $p/n \to γ$. 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 $\vartheta_n \to 0$ and $n\vartheta_n \to \infty$; 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.
Comments74 pages (25-page main text plus supplementary material), 1 figure
