Multivariate analysis and Jacobi ensembles: largest eigenvalue, Tracy-Widom limits and rates of convergence. (English) Zbl 1284.62320

Johnstone, Iain M.

Ann. Stat. 36, No. 6, 2638-2716 (2008).

Summary: 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.

MSC:

62H10	Multivariate distribution of statistics
62E20	Asymptotic distribution theory in statistics
60B20	Random matrices (probabilistic aspects)

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

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