{"ID":2850418,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.21077","arxiv_id":"2510.21077","title":"Limiting Spectral Distribution of High-dimensional Multivariate Kendall-$τ$","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 $\\frac{1}{2}pK_n$ 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 $\\frac{1}{2}trΣK_n$. The theoretical findings are validated through comprehensive simulation studies.","short_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 $\\frac{1}{2}pK_n$ converges almost surely to the Marčenko--Pastur law wi...","url_abs":"https://arxiv.org/abs/2510.21077","url_pdf":"https://arxiv.org/pdf/2510.21077v3","authors":"[\"Ruoyu Wu\"]","published":"2025-10-24T01:25:48Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"math.PR\"]","methods":"[]","has_code":false}