Accuracy of the Tracy-Widom limit for the largest eigenvalue in white Wishart matrices

TL;DR

This paper demonstrates that with refined centering and scaling, the Tracy-Widom law provides a highly accurate second-order approximation for the distribution of the largest eigenvalue of Wishart matrices, enhancing its practical utility.

Contribution

The authors improve the accuracy of the Tracy-Widom approximation for Wishart matrices by identifying optimal centering and scaling constants, achieving second-order precision.

Findings

Second-order accuracy of O(min(n,p)^{-2/3}) for the Tracy-Widom approximation

Numerical simulations confirm the improved approximation quality

Parallel results established for the smallest eigenvalue when n > p

Abstract

Let A be a p-variate real Wishart matrix on n degrees of freedom with identity covariance. The distribution of the largest eigenvalue in A has important applications in multivariate statistics. Consider the asymptotics when p grows in proportion to n, it is known from Johnstone (2001) that after centering and scaling, these distributions approach the orthogonal Tracy-Widom law for real-valued data, which can be numerically evaluated and tabulated in software. Under the same assumption, we show that more carefully chosen centering and scaling constants improve the accuracy of the distributional approximation by the Tracy-Widom limit to second order: O(min(n,p)^{-2/3}). Together with the numerical simulation, it implies that the Tracy-Widom law is an attractive approximation to the distributions of these largest eigenvalues, which is important for using the asymptotic result in practice. We also provide a parallel accuracy result for the smallest eigenvalue of A when n > p.