Tail copula representation of path-based maximal tail dependence

2026-04-07

AI Summary

This work addresses the limitation of classical tail dependence measures, which rely solely on the copula diagonal and thus fail to capture asymmetric tail dependence. The authors propose a novel framework of path-based maximal tail dependence, establishing—for the first time under a non-degenerate tail copula—the existence of a maximal dependence path and its associated coefficient. They provide an explicit characterization of this path along with a one-dimensional optimization representation. By integrating tail copula theory, extreme value theory, and optimization techniques, the framework substantially enhances both the expressiveness and computational tractability of tail dependence analysis. The approach is successfully applied to derive the asymptotic tail behavior of the t-copula and the Marshall–Olkin survival copula.

Abstract

The classical tail dependence coefficient (TDC) may fail to capture non-exchangeable features of tail dependence due to its restrictive focus on the diagonal of the underlying copula. To address this limitation, the framework of path-based maximal tail dependence has been proposed, where a path of maximal dependence is derived to capture the most pronounced feature of dependence over all possible paths, and the path-based maximal TDC serves as a natural analogue of the classical TDC along this path. However, the theoretical foundations of path-based tail analyses, in particular the existence and analytical tractability, have remained limited. This paper addresses this issue in several ways. First, we prove the existence of a path of maximal dependence and the path-based maximal TDC when the underlying copula admits a non-degenerate tail copula. Second, we obtain an explicit characterization of the maximal TDC in terms of the tail copula. Third, we show that the first-order asymptotics of a path of maximal dependence is characterized by a one-dimensional optimization involving the tail copula. These results improve the analytical and computational tractability of path-based tail analyses. As an application, we derive the asymptotic behavior of a path of maximal dependence for the bivariate t-copula and the survival Marshall--Olkin copula.

Problem

• tail dependence
• copula
• path-based maximal tail dependence
• tail copula
• non-exchangeable dependence

Innovation

• tail copula
• path-based maximal tail dependence
• tail dependence coefficient
• asymptotic analysis
• copula theory