Bootstrapping Empirical Processes of Cluster Functionals with Application to Extremograms

Abstract: In the extreme value analysis of time series, not only the tail behavior is of interest, but also the serial dependence plays a crucial role. Drees and Rootz\'en (2010) established limit theorems for a general class of empirical processes of so-called cluster functionals which can be used to analyse various aspects of the extreme value behavior of mixing time series. However, usually the limit distribution is too complex to enable a direct construction of confidence regions. Therefore, we suggest a multiplier block bootstrap analog to the empirical processes of cluster functionals. It is shown that under virtually the same conditions as used by Drees and Rootz\'en (2010), conditionally on the data, the bootstrap processes converge to the same limit distribution. These general results are applied to construct confidence regions for the empirical extremogram introduced by Davis and Mikosch (2009). In a simulation study, the confidence intervals constructed by our multiplier block bootstrap approach compare favorably to the stationary bootstrap proposed by Davis et al.\ (2012).