The exact region determined by Spearman's rho and Gini's gamma
Abstract
We determine the exact attainable region of Spearman's rho and Gini's gamma over all bivariate copulas, resolving the remaining pairwise exact-region problem among Spearman's rho, Kendall's tau, Gini's gamma, Blomqvist's beta and Spearman's footrule. We give an explicit parametrization of the rho-maximal boundary for prescribed Gini's gamma and construct copulas attaining every boundary point. The boundary consists of an elementary arc up to a single junction and, beyond it, countably many algebraic pieces accumulating at comonotonicity. This description also gives the largest possible value of |ρ−γ| and the sharp thresholds beyond which rho and gamma must have the same sign. Our proof separates the signs and magnitudes of centred ranks, reducing the optimization of linear combinations of ρ and γ to an optimal transport problem for two uniform magnitudes. The extremizers combine a central antidiagonal block with rescaled auxiliary transport optimizers, and global optimality follows from a Kantorovich dual potential obtained by gluing the corresponding dual pieces.
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BibTeX
@misc{ansari2026exact,
title = {{The exact region determined by Spearman's rho and Gini's gamma}},
author = {Jonathan Ansari and Marcus Rockel and Stefanie Steinmaßl},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19890},
eprint = {2609.19890},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19890},
}
RIS
TY - UNPB
TI - The exact region determined by Spearman's rho and Gini's gamma
AU - Jonathan Ansari
AU - Marcus Rockel
AU - Stefanie Steinmaßl
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19890
UR - https://arxiv.org/abs/2609.19890
AB - We determine the exact attainable region of Spearman's rho and Gini's gamma over all bivariate copulas, resolving the remaining pairwise exact-region problem among Spearman's rho, Kendall's tau, Gini's gamma, Blomqvist's beta and Spearman's footrule. We give an explicit parametrization of the rho-maximal boundary for prescribed Gini's gamma and construct copulas attaining every boundary point. The boundary consists of an elementary arc up to a single junction and, beyond it, countably many algebraic pieces accumulating at comonotonicity. This description also gives the largest possible value of |ρ−γ| and the sharp thresholds beyond which rho and gamma must have the same sign. Our proof separates the signs and magnitudes of centred ranks, reducing the optimization of linear combinations of ρ and γ to an optimal transport problem for two uniform magnitudes. The extremizers combine a central antidiagonal block with rescaled auxiliary transport optimizers, and global optimality follows from a Kantorovich dual potential obtained by gluing the corresponding dual pieces.
ER -