
Mathematics 2008
Accuracy of the TracyWidom limit for the largest eigenvalue in white Wishart matricesAbstract: Let A be a pvariate real Wishart matrix on n degrees of freedom with identity covariance. The distribution of the largest eigenvalue in A has important applications in multivariate statistics. Consider the asymptotics when p grows in proportion to n, it is known from Johnstone (2001) that after centering and scaling, these distributions approach the orthogonal TracyWidom law for realvalued data, which can be numerically evaluated and tabulated in software. Under the same assumption, we show that more carefully chosen centering and scaling constants improve the accuracy of the distributional approximation by the TracyWidom limit to second order: O(min(n,p)^{2/3}). Together with the numerical simulation, it implies that the TracyWidom law is an attractive approximation to the distributions of these largest eigenvalues, which is important for using the asymptotic result in practice. We also provide a parallel accuracy result for the smallest eigenvalue of A when n > p.
