  In this paper we consider the extreme behavior of the extremal eigenvalues of
white Wishart matrices, which plays an important role in multivariate analysis.
In particular, we focus on the case when the dimension of the feature p is much
larger than or comparable to the number of observations n, a common situation
in modern data analysis. We provide asymptotic approximations and bounds for
the tail probabilities of the extremal eigenvalues. Moreover, we construct
efficient Monte Carlo simulation algorithms to compute the tail probabilities.
Simulation results show that our method has the best performance amongst known
approximation approaches, and furthermore provides an efficient and accurate
way for evaluating the tail probabilities in practice.
