On the Tracy-Widom approximation of studentized extreme eigenvalues of Wishart matrices

The few largest eigenvalues of Wishart matrices are useful in testing numerous hypotheses and are typically studentized as the noise variance is unknown. Specifically, the largest eigenvalue is studentized using the average trace of the matrix. However, this ratio has a distribution poorly approximated by its asymptotic one when either the sample size or dimension is not too large, making inference problematic. We present a simple variance adjustment that significantly improves the approximation and theoretically demonstrate the increase in power that this adjustment delivers compared to the power of the uncorrected studentized eigenvalue. We propose a bias corrected consistent estimator of the noise variance when studentizing the (k+1)st largest eigenvalue in the presence of exactly k spikes and a variance correction for the resulting studentized eigenvalue is proposed.