Comparing and measuring intermediate tail dependence

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 $Λ\equiv 0$. 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 $C(u w)\sim u^κ\ell(u)\,b(w)$, where $κ$ is the tail order, $\ell$ is slowly varying and $b$ is the tail order function of Hua and Joe. We introduce an asymptotic tail order $\preceq$ that refines the tail dependence order and show that it is lexicographic in the three quantities $κ$, $\ell$ and $b$. As a consequence, every real-valued measure that is monotone with respect to $\preceq$ 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.