Limiting Spectral Distribution of High-dimensional Multivariate Kendall-$\tau$

TL;DR

This paper investigates the spectral distribution of the multivariate Kendall-$\tau$ statistic in high dimensions, showing it converges to the Marčenko–Pastur law and extending results to the independent component model.

Contribution

It establishes the limiting spectral distribution of the Kendall-$\tau$ matrix and extends classical results to high-dimensional robust statistics.

Findings

ESD of $\frac{1}{2}pK_n$ converges to Marčenko–Pastur law

Derived fixed-point equation for $\frac{1}{2}tr\Sigma K_n$

Validated theoretical results with simulations

Abstract

The multivariate Kendall-$\tau$ statistic, denoted by $K_n$, plays a significant role in robust statistical analysis. This paper establishes the limiting properties of the empirical spectral distribution (ESD) of $K_n$. We demonstrate that the ESD of $\frac{1}{2}pK_n$ converges almost surely to the Mar\v{c}enko--Pastur law with variance parameter $\frac{1}{2}$, analogous to the classical result for sample covariance matrices. Using Stieltjes transform techniques, we extend these results to the independent component model, deriving a fixed-point equation that characterizes the limiting spectral distribution of $\frac{1}{2}tr\Sigma K_n$. The theoretical findings are validated through comprehensive simulation studies.