Rank weighting and asymmetry in Blest's rank correlation: two exact regions
Abstract
Blest's rank correlation ν is a variant of Spearman's rho ρ that weights the leading ranks of one variable more heavily, at the price that ν is not symmetric in its arguments. We quantify both features by determining the exact region of (ρ,ν) over all bivariate copulas, as well as that of (η,ν), where η is the symmetrized Blest coefficient of Genest and Plante. The latter region is a linear image of the set of all pairs (ν(C),ν(C⊤)) formed by a copula C and its transpose. Consequently, Blest's coefficient differs from Spearman's rho by at most 1/4, and interchanging the two variables changes it by at most 27/64, improving on the bound 1/2 implied by the first inequality. For every given value of ρ or η, each corresponding extreme value of ν is attained by exactly one copula, given in closed form and supported on finitely many line segments. Near countermonotonicity, the upper extremizers of the (η,ν)-region are supported on the graph of a function of the second coordinate, yet their conditional laws given the first coordinate carry two atoms. The proofs rest on a rearrangement inequality with equality case and on explicit Kantorovich potentials.
Related
- Exact regions for the symmetrized Blest coefficient
- Exact PID and PI Gain Regions for Uncertain Non-Affine MIMO Systems
- The exact dimensional threshold for Spearman rank-correlation compatibility
- CorrFlow: Exact and Permutation-Equivariant Flow Matching on the Correlation Elliptope
- Exponential Seniority Weighting Chain-Ladder Reserving
BibTeX
@misc{rockel2026rank,
title = {{Rank weighting and asymmetry in Blest's rank correlation: two exact regions}},
author = {Marcus Rockel},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27634},
eprint = {2609.27634},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27634},
}
RIS
TY - UNPB
TI - Rank weighting and asymmetry in Blest's rank correlation: two exact regions
AU - Marcus Rockel
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27634
UR - https://arxiv.org/abs/2609.27634
AB - Blest's rank correlation ν is a variant of Spearman's rho ρ that weights the leading ranks of one variable more heavily, at the price that ν is not symmetric in its arguments. We quantify both features by determining the exact region of (ρ,ν) over all bivariate copulas, as well as that of (η,ν), where η is the symmetrized Blest coefficient of Genest and Plante. The latter region is a linear image of the set of all pairs (ν(C),ν(C^⊤)) formed by a copula C and its transpose. Consequently, Blest's coefficient differs from Spearman's rho by at most 1/4, and interchanging the two variables changes it by at most 27/64, improving on the bound 1/2 implied by the first inequality. For every given value of ρ or η, each corresponding extreme value of ν is attained by exactly one copula, given in closed form and supported on finitely many line segments. Near countermonotonicity, the upper extremizers of the (η,ν)-region are supported on the graph of a function of the second coordinate, yet their conditional laws given the first coordinate carry two atoms. The proofs rest on a rearrangement inequality with equality case and on explicit Kantorovich potentials.
ER -