AI 中文总结
本文研究乘子自助法对高维样本协方差矩阵最大特征值渐近分布的影响，揭示其在不同乘子下的边缘相变，验证其用于高维谱推断的可行性与有效性。
AI 中文摘要
本文研究了在尖峰模型和非尖峰模型下，采用乘子自助法分析高维样本协方差矩阵最大特征值渐近分布的效果。研究发现，乘子自助法在无条件和有条件自助协方差矩阵的极限边缘分布中建立了若干相变，前提是存在不同类别的乘子。在非尖峰情形中，无界乘子会导致自助协方差矩阵最大特征值的分布出现Fréchet或Gumbel极限，无论是在观测数据的条件下还是无条件下均如此。对于有界乘子，无条件模型会呈现Tracy-Widom、高斯或Weibull极限之间的相变，该相变由纵横比p/n、乘子的上端点行为以及总体协方差矩阵共同决定。有条件模型显示出类似的高斯和Weibull机制；相比之下，无条件Tracy-Widom机制的有条件对应机制会坍缩为一个点质量。在尖峰情形中，在合适的信号强度条件下，在一些温和假设下，无条件和有条件自助样本协方差矩阵的主导特征值对于有界及无界乘子均渐近服从高斯分布。本文的理论结果还阐明了乘子自助法在高维样本协方差模型谱推断中的可行性和适应性。数值模拟证实了本文结果的准确性以及所提出的谱推断程序的有效性，该程序可能具有独立研究价值。
英文摘要
In this paper, we study the effects of employing multiplier bootstrap to analyze the asymptotic distributions of the largest eigenvalues of high-dimensional sample covariance matrices in both spiked and non-spiked models. Our findings demonstrate that the multiplier bootstrap establishes several phase transitions in the limiting edge distributions of both unconditional and conditional bootstrapped covariance matrices, provided the different classes of multipliers. In the nonspiked setting, unbounded multipliers lead to Frechet or Gumbel limits for the largest eigenvalue of the bootstrapped covariance matrix, both conditionally on the observed data and unconditionally. For bounded multipliers, the unconditional model exhibits transitions among Tracy-Widom, Gaussian, or Weibull limits, determined jointly by the aspect ratio p/n, the upper-endpoint behavior of the multipliers, and the population covariance matrix. The conditional model displays analogous Gaussian and Weibull regimes; in contrast, the conditional counterpart of the unconditional Tracy-Widom regime collapses to a point mass. In the spiked setting, under suitable signal-strength conditions, the leading eigenvalues of both the unconditional and conditional bootstrapped sample covariance matrices are asymptotically Gaussian for bounded as well as unbounded multipliers, under some mild assumptions. Our theoretical results also clarify the feasibility and adaptability of the multiplier bootstrap for spectral inference in high-dimensional sample covariance models. Numerical simulations confirm the accuracy of our results and the effectiveness of the proposed spectral inference procedures, which may be of independent interest.
Comments114 pages, 3 tables, 3 figures
