标题： 高维多变量Kendall-$τ$的极限谱分布 显示英文标题

标题： Limiting Spectral Distribution of High-dimensional Multivariate Kendall-$τ$

摘要： 多元Kendall-$\tau$统计量，记为$K_n$，在稳健统计分析中起着重要作用。 本文建立了$K_n$的经验谱分布（ESD）的极限性质。 我们证明了$\frac{1}{2}pK_n$的ESD几乎必然收敛到方差参数为$\frac{1}{2}$的Marčenko--Pastur定律，类似于样本协方差矩阵的经典结果。 利用Stieltjes变换技术，我们将这些结果扩展到独立分量模型，推导出一个固定点方程，该方程表征了$\frac{1}{2}tr\Sigma K_n$的极限谱分布。 理论结果通过全面的模拟研究得到了验证。 显示英文摘要

摘要： 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.