Rare-event Analysis for Extremal Eigenvalues of white Wishart matrices

TL;DR

This paper investigates the tail behavior of the largest and smallest eigenvalues of white Wishart matrices in high-dimensional settings, providing asymptotic bounds and efficient simulation methods for practical evaluation.

Contribution

It introduces new asymptotic approximations and Monte Carlo algorithms for tail probabilities of extremal eigenvalues in high-dimensional Wishart matrices.

Findings

Monte Carlo algorithms outperform existing methods

Accurate tail probability estimates in high-dimensional regimes

Improved bounds for extremal eigenvalues

Abstract

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.