AI 中文总结
本文研究随机缺失观测下高维Gram矩阵的谱特性，证明其经验谱分布收敛性与线性谱统计量的中心极限定理，发现缺失场景下特征值统计量波动受总体协方差特征向量影响的新现象，并将结果应用于协方差矩阵假设检验。
AI 中文摘要
受随机缺失观测场景下利用Gram矩阵进行统计推断的需求驱动，本文研究了随机矩阵$\bold S_n=\frac{1}{n}\bold Z\bold Z^*$的谱特性，其中$\bold Z=\bold D\boldsymbol{\bigcirc}(\boldsymbol{\bold Σ^{1/2}}\bold X)$是Hadamard随机矩阵，其元素由独立伯努利变量$\bold D$决定。在高维框架下，我们证明了$\bold S_n$的经验谱分布收敛于一个明确定义的极限分布。此外，我们探究了缺失机制对Gram矩阵$\bold S_n$谱分布二阶性质的影响，建立了$\bold S_n$线性谱统计量的中心极限定理，揭示了其波动规律。令人意外的是，我们的分析表明，即便在理想高斯分布场景下，随机缺失情形中由特征值生成的统计量的波动仍受总体协方差矩阵的特征向量影响。这一发现揭示了一种与经典情形形成鲜明对比的显著现象。随后，我们展示了该中心极限定理在总体协方差矩阵假设检验中的实际应用。
英文摘要
Motivated by the statistical inference using the Gram matrix in the context of missing at random observations, this paper investigates the spectral properties of the random matrices $\mb S_n=\frac{1}{n}\mb Z\mb Z^*$, where $\mb Z=\mb D\circ(\boldsymbol{Σ^{1/2}}\mb X)$ represents a Hadamard random matrix with entries determined by independent Bernoulli variables $\mb D$. Operating within the high-dimensional framework, we establish the convergence of the empirical spectral distribution of $\mb S_n$ to a well-defined limiting distribution. In addition, we explore the impact of the missing mechanism on the second-order properties of the spectral distribution of the Gram matrix $\mathbf{S}_n$. We establish the central limit theorem for the linear spectral statistics of $\mathbf{S}_n$, shedding light on their fluctuations. Surprisingly, our analysis reveals that even in the ideal Gaussian distribution scenario, the fluctuations of statistics generated by eigenvalues are influenced by the eigenvectors of the population covariance matrix in the missing-at-random case. This discovery uncovers a remarkable phenomenon that starkly contrasts with the classical case. Subsequently, we demonstrate the practical application of our central limit theorem in hypothesis testing for the population covariance matrix.
Journal refAnn. Statist. 52(3): 1254-1275 (June 2024)
