Models with Hidden Regular Variation: Generation and Detection

Abstract

We review the notions of multivariate regular variation (MRV) and hidden regular variation (HRV) for distributions of random vectors and then discuss methods for generating models exhibiting both properties concentrating on the non-negative orthant in dimension two. Furthermore we suggest diagnostic techniques that detect these properties in multivariate data and indicate when models exhibiting both MRV and HRV are plausible fits for the data. We illustrate our techniques on simulated data, as well as two real Internet data sets.1. Introduction. This paper discusses methods for constructing multivariate heavy-tailed models on particular sub-cones of R 2 + = [0, ∞) 2 by adapting techniques from multivariate regular variation theory. We evaluate two different methods for generating models exhibiting multiple heavy-tailed regimes. The results obtained in the different regimes are governed by the sub-cone that serves as the state-space, the choice of scaling function, and often the interaction between different regimes. We also discuss statistical detection methods which validate that data is consistent with particular multivariate heavy-tailed models; these methods are adapted from those developed for the conditional extreme value model (CEV) [8]. We discuss multivariate regular variation (MRV) on the cones R 2 + \ {0} and (0, ∞) 2 . When regular variation exists on both cones, the regular variation on the smaller cone (0, ∞) 2 is called hidden regular variation (HRV).Data that may be modeled by distributions having heavy tails appear in many contexts, for example, hydrology [1], finance [31], insurance [14], Internet traffic [6], social networks and random graphs [3,13,28,30] and risk