It is well known that most of the existing theoretical results in statistics are based on the assumption that the sample is generated with replacement from an infinite population. However, in practice, available samples are almost always collected without replacement. If the population is a finite set of real numbers, whether we can still safely use the results from samples drawn without replacement becomes an important problem. In this paper, we focus on the eigenvalues of high-dimensional sample covariance matrices generated without replacement from finite populations. Specifically, we derive the Tracy-Widom laws for their largest eigenvalues and apply these results to parallel analysis. We provide new insight into the permutation methods proposed by Buja and Eyuboglu in [Multivar Behav Res. 27(4) (1992) 509--540]. Simulation and real data studies are conducted to demonstrate our results.

翻译：众所周知，统计学中现有的理论结果大多基于从无限总体中有放回抽样的假设。然而在实践中，可用样本几乎总是以无放回方式收集。当总体为有限实数集时，能否安全运用基于无放回样本的统计结论成为一个重要问题。本文聚焦于从有限总体无放回生成的高维样本协方差矩阵的特征值，具体推导出其最大特征值的Tracy-Widom律，并将这些结果应用于平行分析。我们对Buja与Eyuboglu在[Multivar Behav Res. 27(4) (1992) 509--540]中提出的置换方法提出了新见解。通过模拟实验和真实数据研究验证了本文结论。