多元分析与雅可比系综：最大特征值、特雷西 - 威多姆极限与收敛速率
MULTIVARIATE ANALYSIS AND JACOBI ENSEMBLES: LARGEST EIGENVALUE, TRACY-WIDOM LIMITS AND RATES OF CONVERGENCE.
作者信息
Johnstone Iain M
机构信息
Department of Statistics, Sequoia Hall, 390 Serra Mall, Stanford University, Stanford, California 94305-4065, E-mail:
出版信息
Ann Stat. 2008 Dec 1;36(6):2638. doi: 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.
设(A)和(B)是(p)个变量中的独立中心威沙特矩阵，具有共同协方差，分别具有(m)和(n)个自由度。((A + B)^{-1}B)的最大特征值的分布在多元统计中有许多应用，但精确计算很困难。假设(m)和(n)与(p)成比例增长。我们表明，经过中心化和缩放后，该分布以二阶近似(O(p^{-2/3}))由特雷西 - 威多姆定律逼近。通过使用随机矩阵理论的方法研究雅可比酉系综和正交系综的最大特征值，对于复值数据和实值数据都得到了结果。雅可比多项式在最大零点附近的渐近近似起着核心作用。
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