The Annals of Statistics
- Ann. Statist.
- Volume 36, Number 6 (2008), 2638-2716.
Multivariate analysis and Jacobi ensembles: Largest eigenvalue, Tracy–Widom limits and rates of convergence
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.
Ann. Statist., Volume 36, Number 6 (2008), 2638-2716.
First available in Project Euclid: 5 January 2009
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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
Johnstone, Iain M. Multivariate analysis and Jacobi ensembles: Largest eigenvalue, Tracy–Widom limits and rates of convergence. Ann. Statist. 36 (2008), no. 6, 2638--2716. doi:10.1214/08-AOS605. https://projecteuclid.org/euclid.aos/1231165182