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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2507.23316 |
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| _version_ | 1866915418754514944 |
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| author | Fuchs, Sebastian Limbach, Carsten Schürrer, Fabian |
| author_facet | Fuchs, Sebastian Limbach, Carsten Schürrer, Fabian |
| contents | We explore how the classical concordance measures - Kendall's $τ$, Spearman's rank correlation $ρ$, and Spearman's footrule $ϕ$ - relate to Chatterjee's rank correlation $ξ$ when restricted to lower semilinear copulas. First, we provide a complete characterization of the attainable $τ$-$ρ$ region for this class, thus resolving the conjecture in [18]. Building on this result, we then derive the exact $τ$-$ϕ$ and $ϕ$-$ρ$ regions, obtain a closed-form relationship between $ξ$ and $τ$, and establish the exact $τ$-$ξ$ region. In particular, we prove that $ξ$ never exceeds $τ$, $ρ$, or $ϕ$. Our results clarify the relationship between undirected and directed dependence measures and reveal novel insights into the dependence structures that result from lower semilinear copulas. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_23316 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On exact regions between measures of concordance and Chatterjee's rank correlation for lower semilinear copulas Fuchs, Sebastian Limbach, Carsten Schürrer, Fabian Methodology We explore how the classical concordance measures - Kendall's $τ$, Spearman's rank correlation $ρ$, and Spearman's footrule $ϕ$ - relate to Chatterjee's rank correlation $ξ$ when restricted to lower semilinear copulas. First, we provide a complete characterization of the attainable $τ$-$ρ$ region for this class, thus resolving the conjecture in [18]. Building on this result, we then derive the exact $τ$-$ϕ$ and $ϕ$-$ρ$ regions, obtain a closed-form relationship between $ξ$ and $τ$, and establish the exact $τ$-$ξ$ region. In particular, we prove that $ξ$ never exceeds $τ$, $ρ$, or $ϕ$. Our results clarify the relationship between undirected and directed dependence measures and reveal novel insights into the dependence structures that result from lower semilinear copulas. |
| title | On exact regions between measures of concordance and Chatterjee's rank correlation for lower semilinear copulas |
| topic | Methodology |
| url | https://arxiv.org/abs/2507.23316 |