Revisiting the region determined by Spearman's $ρ$ and Spearman's footrule $ϕ$
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911820677120000 |
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| author | Tschimpke, Marco Schreyer, Manuela Trutschnig, Wolfgang |
| author_facet | Tschimpke, Marco Schreyer, Manuela Trutschnig, Wolfgang |
| contents | Kokol and Stopar ($2023$) recently studied the exact region $Ω_{ϕ,ρ}$ determined by Spearman's footrule $ϕ$ and Spearman's $ρ$ and derived a sharp lower, as well as a non-sharp upper bound for $ρ$ given $ϕ$. Considering that the proofs for establishing these inequalities are novel and interesting, but technically quite involved we here provide alternative simpler proofs mainly building upon shuffles, symmetry, denseness and mass shifting. As a by-product of these proofs we derive several additional results on shuffle rearrangements and the interplay between diagonal copulas and shuffles which are of independent interest. Moreover we finally show that we can get closer to the (non-sharp) upper bound than established in the literature so far. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_00398 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Revisiting the region determined by Spearman's $ρ$ and Spearman's footrule $ϕ$ Tschimpke, Marco Schreyer, Manuela Trutschnig, Wolfgang Statistics Theory 62H20 Kokol and Stopar ($2023$) recently studied the exact region $Ω_{ϕ,ρ}$ determined by Spearman's footrule $ϕ$ and Spearman's $ρ$ and derived a sharp lower, as well as a non-sharp upper bound for $ρ$ given $ϕ$. Considering that the proofs for establishing these inequalities are novel and interesting, but technically quite involved we here provide alternative simpler proofs mainly building upon shuffles, symmetry, denseness and mass shifting. As a by-product of these proofs we derive several additional results on shuffle rearrangements and the interplay between diagonal copulas and shuffles which are of independent interest. Moreover we finally show that we can get closer to the (non-sharp) upper bound than established in the literature so far. |
| title | Revisiting the region determined by Spearman's $ρ$ and Spearman's footrule $ϕ$ |
| topic | Statistics Theory 62H20 |
| url | https://arxiv.org/abs/2404.00398 |