  Describing the complex dependence structure of extreme phenomena is
particularly challenging. To tackle this issue we develop a novel statistical
algorithm that describes extremal dependence taking advantage of the inherent
hierarchical dependence structure of the max-stable nested logistic
distribution and that identifies possible clusters of extreme variables using
reversible jump Markov chain Monte Carlo techniques. Parsimonious
representations are achieved when clusters of extreme variables are found to be
completely independent. Moreover, we significantly decrease the computational
complexity of full likelihood inference by deriving a recursive formula for the
nested logistic model likelihood. The algorithm performance is verified through
extensive simulation experiments which also compare different likelihood
procedures. The new methodology is used to investigate the dependence
relationships between extreme concentration of multiple pollutants in
California and how these pollutants are related to extreme weather conditions.
Overall, we show that our approach allows for the representation of complex
extremal dependence structures and has valid applications in multivariate data
analysis, such as air pollution monitoring, where it can guide policymaking.
