Tail copula representation of path-based maximal tail dependence

TL;DR

This paper develops a theoretical framework for path-based maximal tail dependence using tail copulas, proving existence, providing explicit characterizations, and analyzing asymptotics for specific copulas.

Contribution

It establishes the existence and analytical form of path-based maximal tail dependence, enhancing the understanding and computation of tail dependence beyond classical measures.

Findings

Proved the existence of a path of maximal dependence for non-degenerate tail copulas.

Derived explicit characterization of the maximal tail dependence in terms of tail copulas.

Analyzed the asymptotic behavior for bivariate t-copula and survival Marshall--Olkin 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.