Subsampling techniques and the Jackknife methodology in the estimation of the extremal index

Abstract

For a sequence of independent, identically distributed random variables any limiting point process for the time normalized exceedances of high levels is a Poisson process. However, for stationary dependent sequences, under general local and asymptotic dependence restrictions, any limiting point process for the time normalized exceedances of high levels is a compound Poisson process, i.e., there is a clustering of high exceedances, where the underlying Poisson points represent cluster positions, and the multiplicities correspond to the cluster sizes. For such classes of stationary sequences there exists the extremal index theta, 0 <=theta <= 1, directly related to the clustering of exceedances of high values. The extremal index theta is equal to one for independent, identically distributed sequences, i.e., high exceedances appear individually, and theta>0 for "almost all" cases of interest. The estimation of the extremal index through the use of the Generalized Jackknife methodology, possibly together with the use of subsampling techniques, is performed. Case studies in the fields of environment and finance will illustrate the performance of the new extremal index estimator comparatively to the classical one. (C) 2007 Elsevier B.V. All rights reserved.