Accuracy of the Tracy--Widom limits for the extreme eigenvalues in white Wishart matrices

TL;DR

This paper evaluates the accuracy of Tracy--Widom law approximations for the extreme eigenvalues of Wishart matrices, demonstrating high precision even for small matrix dimensions through theoretical analysis and numerical validation.

Contribution

It provides a detailed analysis of the accuracy of Tracy--Widom limits for extreme eigenvalues in Wishart matrices, including new error bounds and practical insights for finite sample sizes.

Findings

Approximation accuracy is of order O(min(n,p)^{-2/3})

Log transform improves accuracy for smallest eigenvalue when γ>1

Numerical results show over 50% and 75% error reduction in typical settings

Abstract

The distributions of the largest and the smallest eigenvalues of a $p$-variate sample covariance matrix $S$ are of great importance in statistics. Focusing on the null case where $nS$ follows the standard Wishart distribution $W_p(I,n)$, we study the accuracy of their scaling limits under the setting: $n/p\rightarrow \gamma\in(0,\infty)$ as $n\rightarrow \infty$. The limits here are the orthogonal Tracy--Widom law and its reflection about the origin. With carefully chosen rescaling constants, the approximation to the rescaled largest eigenvalue distribution by the limit attains accuracy of order ${\mathrm {O}({\min(n,p)^{-2/3}})}$. If $\gamma>1$, the same order of accuracy is obtained for the smallest eigenvalue after incorporating an additional log transform. Numerical results show that the relative error of approximation at conventional significance levels is reduced by over 50% in rectangular and over 75% in `thin' data matrix settings, even with $\min(n,p)$ as small as 2.