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Multivariate Analysis and Jacobi Ensembles: Largest eigenvalue, Tracy Widom Limits and Rates of Convergence | | Iain M. Johnstone
; | | Date: |
23 Mar 2008 | | Abstract: | 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. | | Source: | arXiv, 0803.3408 | | Services: | Forum | Review | PDF | Favorites | |
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