Estimating tail probabilities of the ratio of the largest eigenvalue to the trace of a Wishart matrix

This paper develops an efficient Monte Carlo method to estimate the tail probabilities of the ratio of the largest eigenvalue to the trace of the Wishart matrix, which plays an important role in multivariate data analysis. The estimator is constructed based on a change-of-measure technique and it is proved to be asymptotically efficient for both the real and complex Wishart matrices. Simulation studies further show the improved performance of the proposed method over existing approaches based on asymptotic approximations, especially when estimating probabilities of rare events.