An Alternative Characterization of Hidden Regular Variation in Joint Tail Modeling

In modeling the joint upper tail of a multivariate distribution, a fundamental deficiency of classical extreme value theory is the inability to distinguish between asymptotic independence and exact independence. In this work, we examine multivariate threshold modeling based on the framework of regular variation on cones. Tail dependence is described by an angular measure, which in some cases is degenerate on joint tail regions despite strong sub-asymptotic dependence in such regions. The canonical example is a bivariate Gaussian distribution with any correlation less than one. Hidden regular variation (Resnick, 2002), a second-order tail decay on these regions, offers a refinement of the classical theory. Previous characterizations of random vectors with hidden regular variation are not well-suited for joint tail estimation in finite samples, and estimation approaches thus far have been unable to model both the heavier-tailed regular variation and the hidden regular variation simultaneously. We propose to represent a random vector with hidden regular variation as the sum of independent firstand second-order regular varying pieces. We show our model is asymptotically valid via the concept of multivariate tail equivalence, and illustrate simulation methods with the bivariate Gaussian example. Finally, we outline a framework for estimation from our model via the EM algorithm.