The exact region determined by Blomqvist's beta, Spearman's footrule and Kendall's tau
Abstract
We determine the exact region Ωβ,ϕ,τ= {(β(C),ϕ(C),τ(C))| C∈𝒞} of possible joint values of Blomqvist's beta, Spearman's footrule and Kendall's tau over the class 𝒞 of all bivariate copulas. The region consists precisely of all triples (β,ϕ,τ) satisfying −1≤ β≤ 1, 3/16(1+β)2−1/2≤ ϕ≤ 1−3/8(1−β)2 and 4/3 ϕ−1/3≤ τ≤ 2/3 ϕ+1/3. In other words, the known exact (β,ϕ)- and (ϕ,τ)-regions already characterize the joint region, so that, once the value of Spearman's footrule is fixed, Blomqvist's beta imposes no additional restriction on the possible values of Kendall's tau. The proof is constructive: two $2$-parameter families of shuffles of M realize all the points in Ωβ,ϕ,τ with ϕ= 3/16(1+β)2−1/2, and ordinal sums spread realizations of these two families over the whole region Ωβ,ϕ,τ.
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BibTeX
@misc{tratar2026exact,
title = {{The exact region determined by Blomqvist's beta, Spearman's footrule and Kendall's tau}},
author = {Liljana Ferbar Tratar and Damjana Kokol Bukovšek and Blaž Mojškerc and Jacob Israel Orenday Lares and Marcus Rockel and Nik Stopar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.12841},
eprint = {2607.12841},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.12841},
}
RIS
TY - UNPB
TI - The exact region determined by Blomqvist's beta, Spearman's footrule and Kendall's tau
AU - Liljana Ferbar Tratar
AU - Damjana Kokol Bukovšek
AU - Blaž Mojškerc
AU - Jacob Israel Orenday Lares
AU - Marcus Rockel
AU - Nik Stopar
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2607.12841
UR - https://arxiv.org/abs/2607.12841
AB - We determine the exact region Ω_β,ϕ,τ= {(β(C),ϕ(C),τ(C))| C∈𝒞} of possible joint values of Blomqvist's beta, Spearman's footrule and Kendall's tau over the class 𝒞 of all bivariate copulas. The region consists precisely of all triples (β,ϕ,τ) satisfying −1≤ β≤ 1, 3/16(1+β)^2−1/2≤ ϕ≤ 1−3/8(1−β)^2 and 4/3 ϕ−1/3≤ τ≤ 2/3 ϕ+1/3. In other words, the known exact (β,ϕ)- and (ϕ,τ)-regions already characterize the joint region, so that, once the value of Spearman's footrule is fixed, Blomqvist's beta imposes no additional restriction on the possible values of Kendall's tau. The proof is constructive: two $2$-parameter families of shuffles of M realize all the points in Ω_β,ϕ,τ with ϕ= 3/16(1+β)^2−1/2, and ordinal sums spread realizations of these two families over the whole region Ω_β,ϕ,τ.
ER -