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

Abstract

The ratio of the largest eigenvalue divided by the trace of a $p\times p$ 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 $p,n\to \infty$ with $p/n\to c$. 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 $p,n$.