Properties

HACs provide a powerful and systematic approach for capturing complex dependence patterns by decomposing the joint distribution into multiple levels of dependence. This unique characteristic empowers researchers and practitioners to gain deeper insights into the relationships among variables, making HACs valuable tools for various applications such as risk management, portfolio optimization, and simulation studies. In this chapter, we explore the fundamental properties of HACs and shed light on their key characteristics. First, in Sect. 4.1, we delve into an important relationship between an HAC and its bivariate margins, which lays the foundation for understanding higher dimensional structures. Next, we investigate a novel method for uniquely decomposing an HAC structure into a set of trivariate HAC structures in Sect. 4.2. It is shown that this decomposition can be exploited for HAC estimation. Moreover, we highlight that the entire structure of an HAC can be recovered solely from its matrix of pairwise Kendall’s \(\tau \) in Sect. 4.3. This finding demonstrates the richness of the information contained in the Kendall’s \(\tau \) matrix and its practical implications for HAC analysis. Finally, we discuss other essential probabilistic and statistical features for practitioners in Sect. 4.4. These include the probabilistic ordering of the HAC, its significance in extreme value theory, and methods to condense the information of the multivariate HAC-distributed variables into a single dimension.