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Spearman–Gini exact-region problem

Let Π\Pi be the set of all bivariate copulas. For C∈ΠC\in\Pi, define Spearman's rho and Gini's gamma by ρ(C)=12∫01∫01C(u,v) du dv−3\rho(C)=12\int_0^1\int_0^1 C(u,v)\,du\,dv-3 and γ(C)=4∫01(C(u,u)+C(u,1−u)−u) du\gamma(C)=4\int_0^1\bigl(C(u,u)+C(u,1-u)-u\bigr)\,du. Determine exactly the attainable region R={(ρ(C),γ(C)):C∈Π}⊆[−1,1]2\mathcal R=\{(\rho(C),\gamma(C)):C\in\Pi\}\subseteq[-1,1]^2, including an explicit description of its boundary and copulas attaining every boundary point.

References

Primary source

arXiv

Additional references

optimal-transportprobabilitystatistics
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Claimed solved

An unrefereed preprint claims to determine the full relationship between Spearman’s rho and Gini’s gamma, but independent verification is not yet recorded.

The problem asks for the exact attainable region of two classical dependence measures, Spearman’s rho and Gini’s gamma. The latest reported work claims a complete boundary description together with extremizers and sharp thresholds.

September 17, 2026 preprint

Jonathan Ansari, Marcus Rockel, and Stefanie Steinmaßl report an explicit parametrization of the boundary, constructions attaining it, and sharp sign and absolute-difference thresholds. If correct, this resolves the remaining pairwise exact-region problem, but the source is an unrefereed preprint and the claim is unverified.

Current status (as of September 2026): A complete solution is claimed in an unrefereed preprint, while its correctness and publication-level verification remain open.

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