Estimation of Cluster Functionals for Regularly Varying Time Series

Abstract

The classical Extreme Value Theory deals with independent random variables. If random variables are dependent, large values tend to cluster (that is, one large value is followed by a series of large values). It is of interest to describe probabilistically the clustering and estimate the relevant cluster functionals. We consider disjoint blocks, sliding blocks and runs estimators of cluster indices. Using a modern theory of multivariate, regularly varying time series, we obtain consistency results and central limit theorems under conditions that can be easily verified for a large class of short-range dependent models. In particular, we show that in the Peak-over-Threshold framework, all the estimators have the same limiting variances. This solves a longstanding open problem and is in contrast to the Block Maxima method. Our findings are illustrated by simulation experiments.