Properties of extremal dependence models built on bivariate max-linearity

Bivariate max-linear models provide a core building block for characterizing bivariate max-stable distributions. The limiting distribution of marginally normalized component-wise maxima of bivariate max-linear models can be dependent (asymptotically dependent) or independent (asymptotically independent). However, for modeling bivariate extremes they have weaknesses in that they are exactly max-stable with no penultimate form of convergence to asymptotic dependence, and asymptotic independence arises if and only if the bivariate max-linear model is independent. In this work we present more realistic structures for describing bivariate extremes. We show that these models are built on bivariate max-linearity but are much more general. In particular, we present models that are dependent but asymptotically independent and others that are asymptotically dependent but have penultimate forms. We characterize the limiting behavior of these models using two new different angular measures in a radial-angular representation that reveal more structure than existing measures.