Extremal index blocks estimator: the threshold and the block size choice

摘要

The main objective of Statistics of Extremes is the estimation of probabilities of rare events. When extending the analysis of the limiting behaviour of the extreme values from independent and identically distributed sequences to stationary sequences a key parameter appears, the extremal index theta, whose accurate estimation is not easy. Here we focus on the estimation of theta using blocks estimators, that can be constructed by using disjoint or sliding blocks. The asymptotic properties for both procedures were studied and compared but both blocks estimators require the choice of a threshold and a block length. Some criteria have appeared for the choice of those nuisance quantities but some research is still needed. We will show how the threshold and the block size choices can affect the estimates. However the main objective of this work is to revisit another estimation procedure that only depends on the block length, although some conditions on the underlying process need to be verified. The associated estimator presents nice asymptotic properties, and for finite samples is here illustrated a stability criterion for choosing the block length and then obtaining the theta estimate. A large simulation study has been performed and an application to daily mean flow discharge rate in the hydrometric station of Fragas da Torre in river Paiva, data collected from 1 October 1946 to 30 April 2012 is done.