Directional Dependence of Extreme Events

Abstract

This paper introduces a novel measure to quantify the directional dependence of extreme events between two variables. The proposed approach is designed to capture asymmetric tail dependence by studying conditional tail expectations of rank-transformed variables, thereby quantifying the behavior of one variable when the other takes extreme values. We investigate the theoretical asymptotic behavior of the associated estimator. The effectiveness of the approach is demonstrated through an extensive simulation study. In addition, we discuss the use of the proposed coefficient for the detection of causal effects in extreme events. Finally, we apply the method to an oceanographic dataset, where the results highlight the strong asymmetric nature of extreme events and identify the dominant directions of extremal influence among key oceanographic variables. As a directional measure of tail dependence, our approach provides a natural tool for exploring causal-effect relationships in extreme-value settings.

Figures (9)

• Figure 1: Asymptotic variance $\sigma^2_C(v)/n$ (dashed curve), squared bias (grey curve), and quadratic risk (black curve) as functions of the threshold $v$. Top graphs correspond to the Clayton copula with $\theta = 0.2$ for $n = 100$ (left) and $n = 1{,}000$ (right). Bottom graphs correspond to the FGM copula with $\theta = 1$ for $n = 100$ (left) and $n = 1{,}000$ (right).
• Figure 2: Simulated samples from Khoudraji’s device, of size $n = 5{,}000$, for three types of copulas (one in each row) and three values of $\delta$ (one in each column).
• Figure 3: Tail dependence measures (from left to right $\chi^{Y\rightarrow X}$, $\chi^{X\rightarrow Y}$, and $\chi(XY)(v)$) in the Gaussian-Khoudraji’s device, with parameter $\rho \in\{0.25,0.75\}$ for the Gaussian copula, for four values of $v$, as functions of $\delta$.
• Figure 4: Tail dependence measures (from left to right $\chi^{Y\rightarrow X}$, $\chi^{X\rightarrow Y}$, and $\chi(XY)(v)$) in the Clayton-Khoudraji’s device, with parameter $\theta\in\{1,3\}$ for the Clayton copula, for four values of $v$, as functions of $\delta$.
• Figure 5: Tail dependence measures (from left to right $\chi^{Y\rightarrow X}$, $\chi^{X\rightarrow Y}$, and $\chi(XY)(v)$) in the Frank-Khoudraji’s device, with parameter $\theta\in\{4,10\}$ for the Frank copula, for four values of $v$, as functions of $\delta$.

Theorems & Definitions (10)

• Definition 2.1: Transformed Conditional Tail Expectation
• Proposition 2.1
• Definition 2.2: Positive Quadrant Dependence
• Definition 2.3: Directional Tail Dependence
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• proof
• proof
• proof