On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart matrix

The ratio of the largest eigenvalue divided by the trace of a p x p random Wishart matrix with n degrees of freedom and an identity covariance matrix plays an important role in various hypothesis testing problems both in statistics and in signal processing In this paper we derive an approximate explicit expression for the distribution of this ratio by considering the joint limit as both p, n -> infinity with p/n -> c Our analysis reveals that even though asymptotically in this limit the ratio follows a Tracy-Widom (TW) distribution one of the leading error terms depends on the second derivative of the TW distribution and is non-negligible for practical values of p in particular for determining tail probabilities We thus propose to explicitly include this term in the approximate distribution for the ratio We illustrate empirically using simulations that adding this term to the TW distribution yields a quite accurate expression to the empirical distribution of the ratio even for small values of p, n (C) 2010 Elsevier Inc All rights reserved