Methods for estimating the upcrossings index: improvements ans comparison

Abstract

The upcrossings index $0\leq η\leq 1,$ a measure of the degree of local dependence in the upcrossings of a high level by a stationary process, plays, together with the extremal index $θ,$ an important role in extreme events modelling. For stationary processes, verifying a long range dependence condition, upcrossings of high thresholds in different blocks can be assumed asymptotically independent and therefore blocks estimators for the upcrossings index can be easily constructed using disjoint blocks. In this paper we focus on the estimation of the upcrossings index via the blocks method and properties such as consistency and asymptotic normality are studied. We also enlarge the family of runs estimators of $η$ and provide an empirical way of checking local dependence conditions that control the clustering of upcrossings to improve the estimates obtained with the runs method. We compare the performance of a range of different estimators for $η$ and illustrate the methods using simulated data and financial data.