Asymptotically efficient estimators for tail probabilities of extremals of $\beta$-Jacobi ensembles

TL;DR

This paper develops asymptotically efficient estimators for the tail probabilities of extremal eigenvalues in $eta$-Jacobi ensembles, leveraging large deviation principles to improve understanding of their extreme value behavior.

Contribution

It introduces new estimators based on large deviation rate functions that accurately capture tail probabilities of extremals in $eta$-Jacobi ensembles under various conditions.

Findings

Estimators effectively characterize large deviation behavior of extremals.

Transformations ensure extremals satisfy large deviations for accurate tail probability estimation.

Results improve understanding of extremal eigenvalues in multivariate analysis.

Abstract

In this paper, we consider the tail probabilities of extremals of $\beta$-Jacobi ensemble which plays an important role in multivariate analysis. The key steps in constructing estimators rely on the rate functions of large deviations. Therefore, under specific conditions, we consider stretching and shifting transformations applied to the $\beta$-Jacobi ensemble to ensure that its extremals satisfy the large deviations. The estimator we construct characterize the large deviation behavior and moderate deviation behavior of extremals under different assumptions.