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

2012/09/30 by Marco Chiani · 2 citations
Computer Science · Mathematics · #Applied mathematics #Bayesian Methods and Mixture Models #Distribution (mathematics) #Eigenvalues and eigenvectors #Gaussian #Inverse-Wishart distribution #Mathematical analysis #Mathematics #Multivariate statistics #Physics #Probability distribution #Quantum mechanics #Random Matrices and Applications #Random matrix #Simple (philosophy) #Statistical Methods and Bayesian Inference #Statistical physics #Statistics #Univariate distribution #Wishart distribution #cs.IT #math.IT #math.ST #stat.TH

paper · pdf · doi:10.1016/j.jmva.2014.04.002

published as Journal of Multivariate Analysis, Vol. 129, p. 69-81, 2014 · Journal of Multivariate Analysis (2014)

openalex publication_date 2014/04/15 · arxiv created 2014/04/22 · arxiv updated 2014/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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.

Citations

• High Dimensional Statistical Inference and Random Matrices
• On the distribution of the largest eigenvalue in principal components analysis
• On the largest eigenvalue of Wishart matrices with identity covariance when n, p and p/n tend to infinity
• Correlation Functions, Cluster Functions, and Spacing Distributions for Random Matrices
• Exact Scaling Functions for One-Dimensional Stationary KPZ Growth

Cited by

• On the Distribution of the \(ℓ ^\underline \mathrm th\) Largest Eigenvalue of Spiked Complex Wishart Matrices
• Spectrum of the axion dark sector
• Random matrix [wikipedia]

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• On the distribution of the largest eigenvalue in principal components analysis
• Multivariate analysis and Jacobi ensembles: Largest eigenvalue, Tracy–Widom limits and rates of convergence
• Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population