Geometric criteria for identifying extremal dependence and flexible modeling via additive mixtures

The framework of geometric extremes is based on the convergence of scaled sample clouds onto a limit set, characterized by a gauge function, with the shape of the limit set determining extremal dependence structures. While it is known that a blunt limit set implies asymptotic independence, the absence of bluntness can be linked to both asymptotic dependence and independence. Focusing on the bivariate case, under a truncated gamma modeling assumption with bounded angular density, we show that a ``pointy'' limit set implies asymptotic dependence, thus offering practical geometric criteria for identifying extremal dependence classes. Suitable models for the gauge function offer the ability to capture asymptotically independent or dependent data structures, without requiring prior knowledge of the true extremal dependence structure. The geometric approach thus offers a simple alternative to various parametric copula models that have been developed for this purpose in recent years. We consider two types of additively mixed gauge functions that provide a smooth interpolation between asymptotic dependence and asymptotic independence. We derive their explicit forms, explore their properties, and establish connections to the developed geometric criteria. Through a simulation study, we evaluate the effectiveness of the geometric approach with additively mixed gauge functions, comparing its performance to existing methodologies that account for both asymptotic dependence and asymptotic independence. The methodology is computationally efficient and yields reliable performance across various extremal dependence scenarios.