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BOOTSTRAP AND JACKKNIFE METHODS IN EXTREMAL INDEX ESTIMATION 61
3.3. The Generalized Jackknife methodology. The Generalized Jackknife methodology, Gray and Schucany (1972), has the properties of estimating the bias and the variance of any estimator, leading to the development of estimators with bias and mean squared error often smaller than those of an initial set of estimators.
Using the information on the bias of the extremal index estimator ⇥bUC, in (2.4), Gomes et al. (2008) considered first a generalized jackknife estimator of order 2, based on ⇥bUC computed at the three levels, k, bk/2c+1 and bk/4c+1, where bxc denotes, as usual, the integer part of x. They got the estimator
⇥bGJ ⌘ ⇥bGJ(k):=5⇥bUC([k/2]+1) 2 ⇥bUC([k/4]+1)+⇥bUC(k) . (3.4)
This is an asymptotically unbiased estimator of ✓, in the sense that it can remove the two dominant components of bias referred to in (2.5).
More generally, Gomes et al. (2008) considered the levels k, b kc + 1 and b 2kc + 1, depending on a tuning parameter , 0 < < 1, and the class of estimators,
bGJ( ) ( 2 + 1) ⇥bUC ([ k] + 1) ⇣⇥bUC [ 2k] + 1 + ⇥bUC(k)⌘
⇥ (k) := (1 )2 . (3.5)
Actually ⇥bGJ(1/2)(k) ⌘ ⇥bGJ, given in (3.4). Among the members of the class in (3.5), those authors have been heuristically led to the choice = 1/4. Distri- butional properties of ⇥bGJ(1/4)(k) have been obtained by Gomes et al. (2008) through simulation techniques, see also Gomes et al. (2015).
In this paper, the estimator ⇥bGJ in (3.4) will be considered for illustrating the simulation study.
Remark 3.1. From several studies, Gomes et al. (2008), Neves et al. (2015), Prata Gomes and Neves (2015a, 2015b), some remarks can be pointed out regarding these estimators:
• The Generalized-Jackknife estimator, ⇥bGJ, shows a more stable sim- ulated mean value, near the target value of the parameter but at ex- penses of a very high variance. This does not enable it to outperform the original estimator, regarding MSE at optimal levels.
• MSE(⇥bGJ) is not so sharp as MSE(⇥bUC), suggesting less dependence on the value k for obtaining the estimate of ✓.