Estimating Tail Probabilities of the Ratio of the Largest Eigenvalue to the Trace of a Wishart Matrix

TL;DR

This paper introduces an efficient Monte Carlo method for accurately estimating rare tail probabilities of the ratio of the largest eigenvalue to the trace in Wishart matrices, improving over existing asymptotic approaches.

Contribution

It develops a novel change-of-measure Monte Carlo estimator that is asymptotically efficient for real and complex Wishart matrices, enhancing rare event probability estimation.

Findings

The proposed method outperforms existing asymptotic approaches in simulations.

It is asymptotically efficient for both real and complex Wishart matrices.

Simulation results demonstrate superior accuracy in estimating tail probabilities.

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 outperformance of the proposed method over existing approaches based on asymptotic approximations, especially when estimating probabilities of rare events.