Comparing and measuring intermediate tail dependence
Abstract
The tail dependence function Λ of a bivariate copula, and the tail dependence order based on it, cannot distinguish between copulas whose tail dependence coefficient vanishes; for all of them Λ≡ 0. Such copulas, among them the Gaussian copulas with correlation ρ<1 and the lower tails of all extreme-value copulas other than the comonotonicity copula, are nevertheless ordered by the much finer information contained in their tail expansion C(u w)∼ uκℓ(u) b(w), where κ is the tail order, ℓ is slowly varying and b is the tail order function of Hua and Joe. We introduce an asymptotic tail order ⪯ that refines the tail dependence order and show that it is lexicographic in the three quantities κ, ℓ and b. As a consequence, every real-valued measure that is monotone with respect to ⪯ is constant on all but countably many tail-order classes; meaningful measures of intermediate tail dependence must therefore be defined level-wise, and we construct such measures. Finally we determine where the fine structure resides: if the joint tail is ray-regular on the exponential scale, then b is determined by the tail order and the one-sided derivatives at ω=1/2 of the angular dependence function of Wadsworth and Tawn; it is a (possibly broken) product of powers. If the joint tail is log-concave on the exponential scale, b is squeezed between two such products.
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BibTeX
@misc{siburg2026comparing,
title = {{Comparing and measuring intermediate tail dependence}},
author = {Karl Friedrich Siburg},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35325},
eprint = {2610.35325},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35325},
}
RIS
TY - UNPB
TI - Comparing and measuring intermediate tail dependence
AU - Karl Friedrich Siburg
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35325
UR - https://arxiv.org/abs/2610.35325
AB - The tail dependence function Λ of a bivariate copula, and the tail dependence order based on it, cannot distinguish between copulas whose tail dependence coefficient vanishes; for all of them Λ≡ 0. Such copulas, among them the Gaussian copulas with correlation ρ<1 and the lower tails of all extreme-value copulas other than the comonotonicity copula, are nevertheless ordered by the much finer information contained in their tail expansion C(u w)∼ u^κℓ(u) b(w), where κ is the tail order, ℓ is slowly varying and b is the tail order function of Hua and Joe. We introduce an asymptotic tail order ⪯ that refines the tail dependence order and show that it is lexicographic in the three quantities κ, ℓ and b. As a consequence, every real-valued measure that is monotone with respect to ⪯ is constant on all but countably many tail-order classes; meaningful measures of intermediate tail dependence must therefore be defined level-wise, and we construct such measures. Finally we determine where the fine structure resides: if the joint tail is ray-regular on the exponential scale, then b is determined by the tail order and the one-sided derivatives at ω=1/2 of the angular dependence function of Wadsworth and Tawn; it is a (possibly broken) product of powers. If the joint tail is log-concave on the exponential scale, b is squeezed between two such products.
ER -