Measuring multivariate maximal tail dependence

2026-05-25

AI Summary

Classical tail dependence coefficients struggle to capture high-dimensional, asymmetric extremal dependence structures. This work proposes the Multivariate Tail Co-Measure (MTCM), which extends bivariate maximal tail dependence to the multivariate setting for the first time. By maximizing tail probabilities over the unit hypercube, MTCM identifies both the direction and strength of the strongest extremal dependence, with rigorous proof establishing the existence of an optimal direction. Leveraging copula theory, closed-form expressions for MTCM are derived for several prominent copula families, including those based on tail copulas, regularly varying Archimedean generators, and the survival Marshall–Olkin model. An empirical application to trivariate annual maximum sea level data from England successfully uncovers off-diagonal extremal risk patterns that conventional methods fail to detect.

Abstract

The classical tail dependence coefficient (TDC) may fail to capture non-exchangeable features of bivariate tail dependence since it evaluates the underlying copula only along the diagonal. To address this limitation, several measures of strongest manifestation of tail dependence have been proposed in the bivariate case, including a measure based on the tail copula of the underlying bivariate copula. This paper introduces and investigates the multivariate maximal tail concordance measure (MTCM) which extends the bivariate measure to the multivariate case. The MTCM quantifies the largest tail mass over lower hyperrectangles of common unit volume, while the associated maximizer identifies the direction of maximal tail probability. We establish fundamental properties of the MTCM in the multivariate case, including existence of an optimal direction. We also derive analytical representations for several important model classes. Closed-form expressions are further obtained for survival Marshall-Olkin copulas, Archimax and nested Archimedean copulas with regularly varying Archimedean generators. An application to trivariate annual sea-level maxima in England shows that the MTCM can reveal off-diagonal stress directions and substantial differences in the underlying extremal dependence not detected by likelihood- or TDC-based comparisons.

Problem

• tail dependence
• multivariate extremes
• copula
• maximal tail concordance
• extremal dependence

Innovation

• multivariate maximal tail concordance measure
• tail dependence
• copula
• extremal dependence
• regularly varying generators