Limiting Spectral Distribution of High-dimensional Multivariate Kendall-$τ$
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909918979686400 |
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| author | Wu, Ruoyu |
| author_facet | Wu, Ruoyu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_21077 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Limiting Spectral Distribution of High-dimensional Multivariate Kendall-$τ$ Wu, Ruoyu Statistics Theory Probability 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. |
| title | Limiting Spectral Distribution of High-dimensional Multivariate Kendall-$τ$ |
| topic | Statistics Theory Probability |
| url | https://arxiv.org/abs/2510.21077 |