Mathematics – Statistics Theory
Scientific paper
2008-03-23
Annals of Statistics 2008, Vol. 36, No. 6, 2638-2716
Mathematics
Statistics Theory
Published in at http://dx.doi.org/10.1214/08-AOS605 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Scientific paper
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 $(A+B)^{-1}B$ 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.
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