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BOOTSTRAP AND JACKKNIFE METHODS IN EXTREMAL INDEX ESTIMATION 63
A sketch of the simulation procedure illustrated in Figure 7 is:
• A random sample of size n = 500 was generated from the ARMAX
process with ✓ = (0.1, 0.4, 0.8);
• An initial block size, b = n1/5 was defined, see Section 3.2.
• The NPPI method was applied for estimating the “optimal” block length for resampling, which depends on the value of k.
• The mode of the block size was adopted as the “optimal” block length.
• Finally block bootstrap estimates for the ⇥bUC and ⇥bGJ were obtained.
5. Some concluding remarks
In the field of statistics of extremes, resampling methodologies have revealed themselves as providing very good results to adequately choose the number k of upper order statistics to be taken in the semi-parametric tail index estimation or to obtain more stable path estimates, around the target value.
Resampling blocks in a dependent set-up instead of resampling individual observations was proposed by some authors. However the accuracy of the es- timation strongly depends on the block size. A computational procedure for estimating the “optimal” block length for resampling in the situation of depen- dence was reviewed in this paper. In a first stage it was only considered to deal with the bias of the estimator. A simulation study for estimating the extremal index through two well known estimators, the Up-crossing and the Generalized Jackknife, was conducted.
The variance of the estimators is another characteristic that needs to be incorporated in the bootstrap block size estimation procedure. Other estimators should be compared and procedures for an adaptive choice of the high level need also to be included in the algorithm. Work on these topics is now in progress.
Acknowledgments
I would like to thank the reviewer for the helpful and valuable comments and suggestions that greatly improved the first version of this paper.
References
[1] J. Beirlant, Y. Goegebeur, J. Segers, and J. L. Teugels, Statistics of Extremes: Theory and Applications, England, John Wiley & Sons, 2004.