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BOOTSTRAP AND JACKKNIFE METHODS IN EXTREMAL INDEX ESTIMATION 51
number of upper order statistics used in the estimation, and a high bias for large k.
Resampling methodologies have provided very fruitful results in the field of statistics of extremes. The bootstrap methodology has been used, in partic- ular, in the choice of the number k of upper order statistics to be taken in the semi-parametric tail index estimation, see Draisma et al. (1999), Danielson et al. (2001), Gomes and Oliveira (2001), Gomes et al. (2012, 2013), Caeiro and Gomes (2014) and Gomes et al. (in press), to mention a few works on boot- strap in statistics of extremes. Bootstrapped version of the estimators usually shows a more stable path which helps the estimation of the optimal sample fraction, through some stability criterion, see for example Gomes and Pestana (2007). The jackknife methodology allows us to obtain the bias and the variance of a given statistic and then to build estimators with bias and Mean Square Error (MSE) smaller than those of initial estimators. Section 2 of this arti- cle is dedicated to define the extremal index and to describe some properties. Some estimators are also presented. Resampling procedures are introduced in Section 3 where attention is specially given to the dependent set-up. Classi- cal bootstrap procedures, derived for independence, have to be adapted to the dependent context. One of the procedures proposed for the bootstrap resam- pling in this situation, see for example Lahiri (2002), consists of defining blocks for resampling, instead of resampling the individual observations. But the per- formance of the bootstrap estimator crucially depends on the block size that must be supplied by the user. So, a computational procedure for estimating that block size is proposed, following Lahiri et al. (2007). A brief illustration of a simulation study that is now in progress is shown in Section 4. The paper finishes with a few comments and some notes regarding the work in progress.
2. The Extremal Index: definition and estimation
The extremal index, ✓, measures the relationship between the dependence structure of the data and the behaviour of the exceedances over a high threshold un. This threshold un is such that, for ⌧ > 0, the underlying d.f. F verifies
F(un) = 1 ⌧/n + o(1/n), n ! 1. (2.1)
A stationary sequence {Xn}n 1 from an underlying model F is said to have an extremal index ✓ (0 < ✓  1) if, for all ⌧ > 0, we can find a sequence of levels un = un(⌧) such that with {Yn}n 1 the associated i.i.d. sequence (i.e., an i.i.d. sequence from the same F ), it is verified
P(Yn:n  un) = Fn(un) ! e ⌧ and P(Xn:n  un) ! e ✓⌧, n!1 n!1
(see Leadbetter et al. (1983)).