Importance Sampling for the Extremal Eigenvalue of $\beta$-Jacobi ensemble

This paper focuses on rare events associated with the tail probabilities of the extremal eigenvalues in the $\beta$-Jacobi ensemble, which plays a critical role in both multivariate statistical analysis and statistical physics. Under the ultra-high dimensional setting, we give an exact approximation for the tail probabilities and construct an efficient estimator for the tail probabilities. Additionally, we conduct a numerical study to evaluate the practical performance of our algorithms. The simulation results demonstrate that our method offers an efficient and accurate approach for evaluating tail probabilities in practice.