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

TL;DR

This paper develops an efficient method to accurately estimate the tail probabilities of extremal eigenvalues in the $eta$-Jacobi ensemble, which is important for high-dimensional statistics and physics.

Contribution

It provides an exact approximation and an efficient estimator for tail probabilities of extremal eigenvalues in the $eta$-Jacobi ensemble under ultra-high dimensional settings.

Findings

The proposed method accurately estimates tail probabilities.

Numerical results confirm efficiency and practicality.

Algorithm outperforms existing approaches in simulations.

Abstract

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.