Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy-Widom distribution

TL;DR

This paper derives exact formulas for the distribution of the largest eigenvalue in real Wishart and GOE matrices and proposes a simple, accurate Gamma distribution approximation for the Tracy-Widom law.

Contribution

It provides recursive formulas for exact distributions and introduces a Gamma approximation for the Tracy-Widom law, simplifying statistical applications.

Findings

Exact recursive formulas for finite-dimensional distributions

Gamma distribution approximates Tracy-Widom law accurately

Efficient computation for practical statistical use

Abstract

We derive efficient recursive formulas giving the exact distribution of the largest eigenvalue for finite dimensional real Wishart matrices and for the Gaussian Orthogonal Ensemble (GOE). In comparing the exact distribution with the limiting distribution of large random matrices, we also found that the Tracy-Widom law can be approximated by a properly scaled and shifted Gamma distribution, with great accuracy for the values of common interest in statistical applications.