缺失观测下带尖峰样本协方差矩阵的极限本征结构
Limiting eigen-structure of spiked sample covariance matrices under missing observations
AI总结：
本文针对带缺失观测的尖峰总体模型，利用随机矩阵理论研究高维PCA的渐近行为，证明其样本特征值的渐近正态性与完整数据情形存在差异，并基于此提出尖峰总体独立结构的检验方法。
AI中文摘要：
高维主成分分析（PCA）已成为现代数据分析的核心工具，可实现降维与特征提取，但缺失数据的存在会带来显著挑战，扭曲PCA的性能并使统计推断复杂化。本文研究带缺失观测的尖峰总体模型下PCA的渐近行为，利用随机矩阵理论的最新进展，证明带尖峰的样本特征值具有渐近正态性，但其极限参数与完整数据情形存在显著差异，反映出缺失数据机制的非平凡影响。作为研究结果的应用，本文提出一项检验，用于评估尖峰总体的独立结构。
英文摘要：
High-dimensional Principal Component Analysis (PCA) has become an essential tool in modern data analysis, offering dimensionality reduction and feature extraction. However, the presence of missing data introduces significant challenges, distorting the performance of PCA and complicating statistical inference. In this paper, we study the asymptotic behavior of PCA under a spiked population model with missing observations, leveraging recent advances in random matrix theory. We demonstrate that while the spiked sample eigenvalues exhibit asymptotic normality, the limiting parameters differ substantially from those in the complete data case, reflecting the non?trivial influence of the missing data mechanism. As an application of our results, we propose a test to evaluate the independent structure of a spiked population.
