  Let A be a p-variate 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 Tracy-Widom
law for real-valued 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 Tracy-Widom limit to second order: O(min(n,p)^{-2/3}). Together with the
numerical simulation, it implies that the Tracy-Widom 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.
