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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | https://arxiv.org/abs/2605.13522 |
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| _version_ | 1866913163866275840 |
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| author | Limbach, Carsten |
| author_facet | Limbach, Carsten |
| contents | We investigate a geometric and distributional reinterpretation of Chatterjee's $ξ$-coefficient, which measures functional dependence between a response variable $Y$ and a predictor vector $\mathbf{X}$. For this purpose, we analyze the Markov product $(Y,Y')$, where $Y'$ is a copy of $Y$ that is conditionally independent of $Y$ given $\mathbf{X}$. Based on this construction, we introduce and study two dependence functions, denoted by $ϕ_{(Y,\mathbf{X})}$ and $κ_{(Y,\mathbf{X})}$.
The proposed framework provides a geometric interpretation of the Markov product and extends Chatterjee's correlation coefficient to a richer and more interpretable object for the analysis of directed stochastic dependence. In particular, rather than only measuring how well $Y$ can be represented as a function of $\mathbf{X}$, the proposed dependence functions additionally quantify how strongly the corresponding Markov product is concentrated near the diagonal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_13522 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dependence functions based on Chatterjee's rank correlation Limbach, Carsten Statistics Theory We investigate a geometric and distributional reinterpretation of Chatterjee's $ξ$-coefficient, which measures functional dependence between a response variable $Y$ and a predictor vector $\mathbf{X}$. For this purpose, we analyze the Markov product $(Y,Y')$, where $Y'$ is a copy of $Y$ that is conditionally independent of $Y$ given $\mathbf{X}$. Based on this construction, we introduce and study two dependence functions, denoted by $ϕ_{(Y,\mathbf{X})}$ and $κ_{(Y,\mathbf{X})}$. The proposed framework provides a geometric interpretation of the Markov product and extends Chatterjee's correlation coefficient to a richer and more interpretable object for the analysis of directed stochastic dependence. In particular, rather than only measuring how well $Y$ can be represented as a function of $\mathbf{X}$, the proposed dependence functions additionally quantify how strongly the corresponding Markov product is concentrated near the diagonal. |
| title | Dependence functions based on Chatterjee's rank correlation |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2605.13522 |