Multivariate and Extreme Value Distributions

Abstract

Up to this point, we have mainly focused our efforts on univariate distributions. This was mostly just to keep the notation simple. Multivariate data, however, appear often in practice and multivariate distributions are eminently useful and important. It is time now to formalise multivariate distributions and explicitly discuss models for multivariate observations. The workhorse in this field is certainly the multivariate normal distribution, which we will explore in depth in Sect. 10.1. Beyond the normal distribution, copula-based distributions have been a hot topic in recent years. Copulas allow for complex dependence structures and this chapter provides a short introduction to the basic ideas of copulas in Sect. 10.2. Besides modelling multivariate data, it often occurs that extreme events are of interest, e.g. a maximal loss or a minimal supply level. This is addressed in Sect. 10.3, where we introduce extreme value distributions.