Limiting Spectral Distribution of High-dimensional Multivariate Kendall-τ

2025/10/24 by Wu, Ruoyu
#FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2510.21077

Abstract

The multivariate Kendall-τ 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 (1)/(2)pK_n converges almost surely to the Marčenko--Pastur law with variance parameter (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 (1)/(2)trΣK_n. The theoretical findings are validated through comprehensive simulation studies.

Citations

• Limiting spectral distribution for large sample correlation matrices
• Large sample correlation matrices: a comparison theorem and its applications
• Limiting spectral distribution of large dimensional Spearman’s rank correlation matrices
• On eigenvalues of a high-dimensional Kendall's rank correlation matrix with dependence
• Tracy-Widom limit for Kendall's tau
• Marchenko-Pastur Law for Tyler's M-estimator
• On the distribution of the largest eigenvalue in principal components analysis

Related

• Marčenko-Pastur Law for Kendall's Tau
• Central Limit Theorem for Linear Spectral Statistics of Large Dimensional Kendall's Rank Correlation Matrices and its Applications
• Tracy-Widom limit for Kendall's tau
• On eigenvalues of a high-dimensional Kendall's rank correlation matrix with dependence
• Another Marčenko-Pastur law for Kendall's tau