Tracy-Widom limit for Kendall's tau

TL;DR

This paper proves that the largest eigenvalue of the Kendall rank correlation matrix follows the Tracy-Widom distribution in high dimensions, marking a first in nonparametric random matrix models and high-dimensional U-statistics.

Contribution

It establishes the Tracy-Widom law for the largest eigenvalue of the Kendall rank correlation matrix, a novel result in nonparametric high-dimensional statistics.

Findings

Tracy-Widom law holds for the Kendall rank correlation matrix.

First Tracy-Widom law for a nonparametric random matrix model.

First Tracy-Widom law for a high-dimensional U-statistic.

Abstract

In this paper, we study a high-dimensional random matrix model from nonparametric statistics called the Kendall rank correlation matrix, which is a natural multivariate extension of the Kendall rank correlation coefficient. We establish the Tracy-Widom law for its largest eigenvalue. It is the first Tracy-Widom law for a nonparametric random matrix model, and also the first Tracy-Widom law for a high-dimensional U-statistic.