Multivariate analysis and Jacobi ensembles: largest eigenvalue, Tracy--Widom limits and rates of convergence

TL;DR

This paper demonstrates that the largest eigenvalue of certain multivariate Wishart matrices converges to the Tracy--Widom distribution at a rate of $O(p^{-2/3})$, using advanced random matrix theory techniques.

Contribution

It provides second-order approximations for the distribution of the largest eigenvalue in Jacobi ensembles, extending Tracy--Widom law applications to multivariate statistics.

Findings

Largest eigenvalue distribution approximates Tracy--Widom law

Convergence rate is $O(p^{-2/3})$

Results apply to both complex and real data

Abstract

Let $A$ and $B$ be independent, central Wishart matrices in $p$ variables with common covariance and having $m$ and $n$ degrees of freedom, respectively. The distribution of the largest eigenvalue of $(A+B)^{-1}B$ has numerous applications in multivariate statistics, but is difficult to calculate exactly. Suppose that $m$ and $n$ grow in proportion to $p$. We show that after centering and scaling, the distribution is approximated to second-order, $O(p^{-2/3})$, by the Tracy--Widom law. The results are obtained for both complex and then real-valued data by using methods of random matrix theory to study the largest eigenvalue of the Jacobi unitary and orthogonal ensembles. Asymptotic approximations of Jacobi polynomials near the largest zero play a central role.