Limiting Spectral Distribution of High-dimensional Multivariate Kendall-$\ddot{I}„$

Abstract

The multivariate Kendall-$\ddot{I}„$ 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 $\frac{1}{2}p{K}_n$ converges almost surely to the Marčenko--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 $\frac{1}{2}tr\hat{I}\pounds K_n$. The theoretical findings are validated through comprehensive simulation studies.