Accuracy of the Tracy-Widom limit for the largest eigenvalue in white Wishart matrices

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.