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December 2008 Multivariate analysis and Jacobi ensembles: Largest eigenvalue, Tracy–Widom limits and rates of convergence
Iain M. Johnstone
Ann. Statist. 36(6): 2638-2716 (December 2008). DOI: 10.1214/08-AOS605


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)−1B 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.


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Iain M. Johnstone. "Multivariate analysis and Jacobi ensembles: Largest eigenvalue, Tracy–Widom limits and rates of convergence." Ann. Statist. 36 (6) 2638 - 2716, December 2008.


Published: December 2008
First available in Project Euclid: 5 January 2009

zbMATH: 1284.62320
MathSciNet: MR2485010
Digital Object Identifier: 10.1214/08-AOS605

Primary: 62H10
Secondary: 15A52 , 62E20

Keywords: canonical correlation analysis , characteristic roots , Fredholm determinant , Jacobi polynomials , largest root , Liouville–Green , multivariate analysis of variance , Random matrix theory , Roy’s test , soft edge , Tracy–Widom distribution

Rights: Copyright © 2008 Institute of Mathematical Statistics


Vol.36 • No. 6 • December 2008
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