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

Abstract

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.

Speeding up Monte Carlo simulations for the adaptive sum of powered score test with importance sampling

“…The proposed distribution is often called a proposal distribution. This method has been frequently used to evaluate the extremes of Gaussian random fields and other rare‐event probabilities, and its efficiency has been carefully studied (Adler et al ., 2012; Liu and Xu, 2014a, 2014b; Jiang et al ., 2017; He and Xu, 2018; Li and Xu, 2018). Since the p ‐value calculation involves sampling from a rejection region with possibly a small probability from a null distribution, we propose using importance sampling to speed up the SPU and aSPU tests.…”