期刊
ANNALS OF STATISTICS
卷 36, 期 6, 页码 2638-2716出版社
INST MATHEMATICAL STATISTICS
DOI: 10.1214/08-AOS605
关键词
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
资金
- NSF [DMS-00-72661, DMS-05-05303]
- NIH [RO1 CA 72028, EB 001988]
Let A and B be independent, central Wishart matrices in p variables with common covariance and having in 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 in 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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