Dependence Structures in Stable Mixture Models with an Application to Extreme Precipitation

Abstract: In this thesis we study a class of mixture models obtained by mixing extreme value distributions over a positive stable distribution. This depicts a group structure, where the stable distribution is a group specific quantity and a function of the surroundings. The stable mixture models possess a number of interesting characteristics. A key feature of these models is that they are extreme value distributed, unconditionally as well as conditionally on the stable variables. Furthermore, all lower dimensional marginals belong to the same class of models. These properties make the models analytically tractable to work with and their applications comprehensible. Finally we have the flexibility quality. We prove that any multivariate extreme value distribution may be approximated by such a model. Because this class of mixture models has a finite parametrization, which in general multivariate extreme value distributions do not have, we now have a finite parametrization for all multivariate extreme value distributions. This means that, given enough complexity, any multivariate extreme value distribution may be described by our stable mixture models. The flexibility of the models enables us to study the dependence structure in a wide range of multivariate extreme value situations. In an environmental context, extreme values at several nearby points in space or time may have profound effects on climate. We present a number of stable mixture models and derive their bivariate dependencies. This gives us a set of models that enable us to study not only the extremal properties of several processes collectively, but also to in a straightforward way describe their inter-relationships. Finally we investigate extreme precipitation patterns in northern Sweden by fitting stable mixture models to annual precipitation maxima. From our results we are able to calculate risks for landslides. Keywords: multivariate extreme value theory, mixture model, stable variable, dependence measure