The exact region between Chatterjee's ξ and Blomqvist's β
Abstract
We determine the exact region of Chatterjee's rank correlation ξ and Blomqvist's β, that is, the set of all pairs $(ξ(C),β(C))$ where C ranges over all bivariate copulas. The region equals {(x,y)∈[0,1]×[−1,1]:|y|3≤2x}, so every copula satisfies the sharp inequality |β(C)|3≤2ξ(C). Its left boundary ξ=|β|3/2 is attained uniquely by an explicit family of copulas that perturb independence by a signed tent function centered at the median, and its right boundary ξ=1 by deterministic interval-exchange copulas. The proof reduces the problem to a one-dimensional minimization of an L2-norm over $1$-Lipschitz functions. We further show that the region does not shrink when restricted to radially symmetric copulas and that positively quadrant dependent copulas attain exactly its upper half. For copulas under which the second variable is stochastically increasing in the first, the exact region is {(x,y)∈[0,1]2:y3≤2x, 3(1−y)2≤4(1−x)}, and its right boundary is attained uniquely by explicit singular two-branch copulas. In particular, under stochastic increasingness β=0 is compatible with every value ξ∈[0,1/4], whereas ξ>1/4 forces β>0.
Related
- The exact region determined by Blomqvist's beta, Spearman's footrule and Kendall's tau
- The monotone part of Chatterjee's rank correlation
- The exact region determined by Spearman's rho and Gini's gamma
- On Exact Critical-Region Boundaries in Continuous-Time Multiparametric Optimal Control Problems
- Sharp convergence rates for mean field control in the region of strong regularity
BibTeX
@misc{lares2026exact,
title = {{The exact region between Chatterjee's $ξ$ and Blomqvist's $β$}},
author = {Jacob Israel Orenday Lares and Marcus Rockel},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.30033},
eprint = {2606.30033},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.30033},
}
RIS
TY - UNPB
TI - The exact region between Chatterjee's ξ and Blomqvist's β
AU - Jacob Israel Orenday Lares
AU - Marcus Rockel
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2606.30033
UR - https://arxiv.org/abs/2606.30033
AB - We determine the exact region of Chatterjee's rank correlation ξ and Blomqvist's β, that is, the set of all pairs $(ξ(C),β(C))$ where C ranges over all bivariate copulas. The region equals {(x,y)∈[0,1]×[−1,1]:|y|^3≤2x}, so every copula satisfies the sharp inequality |β(C)|^3≤2ξ(C). Its left boundary ξ=|β|^3/2 is attained uniquely by an explicit family of copulas that perturb independence by a signed tent function centered at the median, and its right boundary ξ=1 by deterministic interval-exchange copulas. The proof reduces the problem to a one-dimensional minimization of an L^2-norm over $1$-Lipschitz functions. We further show that the region does not shrink when restricted to radially symmetric copulas and that positively quadrant dependent copulas attain exactly its upper half. For copulas under which the second variable is stochastically increasing in the first, the exact region is {(x,y)∈[0,1]^2:y^3≤2x, 3(1−y)^2≤4(1−x)}, and its right boundary is attained uniquely by explicit singular two-branch copulas. In particular, under stochastic increasingness β=0 is compatible with every value ξ∈[0,1/4], whereas ξ>1/4 forces β>0.
ER -