Sampling without replacement from a high-dimensional finite population

TL;DR

This paper investigates the spectral properties of high-dimensional sample covariance matrices drawn without replacement from finite populations, deriving Tracy-Widom laws for their largest eigenvalues and applying these to improve parallel analysis methods.

Contribution

It extends eigenvalue distribution theory to finite populations sampled without replacement, providing new Tracy-Widom laws applicable in high-dimensional settings.

Findings

Derived Tracy-Widom laws for largest eigenvalues without replacement

Applied results to enhance parallel analysis techniques

Validated findings through simulations and real data

Abstract

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.