  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.
