Kenngrößen für die Abhängigkeitsstruktur in Extremwertzeitreihen

Abstract

The extreme value dependence structure in uni- as well as multivariate time series may be described in a simplified way using suitable characteristics. These include concepts such as the multivariate extremal index, the extremal coefficient as well as the extremal coefficient function that are all well-known from the literature. We shall discuss interdependencies between these characteristics that, inter alia, allow for the improvement of the well-known bounds for the multivariate extremal index. In the sequel, following the Herglotz theorem we will discuss a representation for the integer-valued extremal coefficient functions with finite range. This result will, in particular, also allow for the construction of example processes corresponding to given extremal coefficient functions. The corresponding proof is substantially based on a self-contained result, namely the equivalence of integer valued extremal coefficient functions and set covariance functions. The introduced characterization of extremal coefficient functions shows in particular that those extremal dependence structures corresponding e.g. to the homometric sequences known from crystallography may not be distinguished by the extremal coefficient function. As a solution to this problem we propose a modification of the extremal coefficient function that also has a meaningful interpretation in practice. Finally, we consider a way to evaluate the modified extremal coefficient function for GARCH(1,1) processes. To this end, we present an extension of the well-known tail chain approach for ARCH(1) processes.