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56 M. M. NEVES
Figures 3 and 4 illustrate the behaviour of the bias, variance and MSE of the ⇥bUC(k) estimator.
Remark 2.4. Some remarks on Figures 3 and 4:
• The ⇥bUC estimator shows a very strong bias.
• The bias is the dominant component of the MSE.
• MSE(⇥bUC) is very sharp, which reveals a need for a very accurate way of choosing k in order to obtain a reliable estimate of ✓. Alternatively it would be sensible to look for less biased estimators.
Recently resampling techniques have been used, specifically Bootstrap and Jackknife procedures, which have shown themselves to improve the performance of the semi-parametric estimators. Let us refer to some recent works deal- ing with those procedures to estimate parameters in EVT, such as Draisma et al. (1999), Gomes and Oliveira (2001), Danielsson et al. (2001), Gomes et al., (2011, 2012), to cite only a few. Prata Gomes and Neves (2011, 2015a,b) present some results on the use of resampling procedures in the extremal index estimation.
3. Resampling Techniques and the dependent set-up
Resampling methodologies, Efron (1979), have revealed good results in the estimation of the threshold k, and in the reduction of bias of any estimator of a parameter of extreme events. In its classical form, the bootstrap has proven to be a powerful nonparametric tool when based on i.i.d observations. But Singh (1981) showed that it could be inadequate under dependence.
So the bootstrap methodology needs to take into account the two di↵erent situations: resampling from an i.i.d. sequence or resampling from a dependent sequence.
3.1. The Bootstrap under dependence. Several attempts have been made to extend the bootstrap method to the dependent case. A breakthrough was achieved when resampling of single observations was replaced by block resam- pling.
Hall(1985), Carlstein (1986), Ku¨nsch (1989) and Liu and Singh (1992) have independently introduced nonparametric versions of the Bootstrap and Jack- knife applicable to weakly dependent stationary observations. Their resampling technique considers to resample or to delete one-by-one whole blocks of ob- servations to obtain consistent procedures for estimating a parameter of the distribution of the stationary series.