Multivariate analysis and Jacobi ensembles: Largest eigenvalue, Tracy–Widom limits and rates of convergence

2008/03/31 by Iain M. Johnstone · 4 citations
Mathematics · #Advanced Combinatorial Mathematics #Mathematical functions and polynomials #Random Matrices and Applications #math.PR #math.ST #msc:15A52 #msc:62E20 #msc:62H10 #stat.TH

paper · pdf · doi:10.1214/08-aos605

published as Annals of Statistics 2008, Vol. 36, No. 6, 2638-2716 · Published in at http://dx.doi.org/10.1214/08-AOS605 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

openalex publication_date 2008/12/01 · arxiv created 2009/01/21 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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.

Citations

• On the distribution of the largest eigenvalue in principal components analysis

Cited by

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• Limiting distribution of maximal crossing and nesting of Poissonized random matchings
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