Paper: arXiv 2604.05985

Authors: Takaaki Koike, Marius Hofert, Haruki Tsunekawa

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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides strong mathematical proofs and theoretical advancements regarding tail copulas and path-based dependence. However, it is purely theoretical with no empirical backtests or empirical financial data validation included in the provided summary.

Research Flowchart

  flowchart TD
    A[Research Goal: Improve theoretical foundation & tractability of path-based maximal tail dependence] --> B[Input: Bivariate Copulas admitting a non-degenerate Tail Copula]
    B --> C[Method: Formulate 1D optimization problem via Tail Copula representation]
    C --> D[Theoretical Analysis: Prove existence of maximal path & derive explicit TDC characterization]
    D --> E[Application & Computation: Evaluate first-order asymptotics on t-copula & survival Marshall-Olkin copula]
    E --> F[Key Outcome: Proven existence, explicit characterization, & improved computational tractability for maximal TDC]