Page 18 - Textos de Matemática Vol. 47

8 M. I. GOMES
ascending order statistics (OSs) associated with the original sample {Xi}1in, it is thus sensible to consider a deterministic level u 2 [Xn k:n, Xn k+1:n) and
the estimator
ˆN 1 nX 1
✓n (k) := k I[Xj Xn k:n<Xj+1]. (4.1)
j=1
The EI-estimator is then a function of k, the number of OS’s higher than the
chosen threshold. We further assume a sensible structure for the asymptotic bias, given by
Biasn✓ˆN(k)o=' (✓)✓k◆+' (✓)✓1◆+o✓1◆+o✓k◆, (4.2) n1n2kkn
as n ! 1, and for any intermediate k (see Gomes et al., 2008). Indeed, for IID
data (✓ = 1):
Moreover, for ARMAX processes, in {2.3}, we get
E n✓ˆN(k)o = ✓ ✓ ✓(✓ + 1) ✓ k ◆ 3 2 ✓ ◆ (1 + o(1)).
In the semi-parametric EI-estimation we have thus to cope with problems similar to the ones appearing in the EVI-estimation: increasing bias, as the threshold decreases and a high variance for high thresholds. It is then sensible to ask whether it is possible to improve the performance of estimators through the use of resampling methods. We can surely use the generalized jackknife (GJ) methodology, to reduce the bias of the EI-estimators, in (4.1). Indeed, in statistics we often put the question whether the combination of information can improve the quality of estimators of a certain parameter or functional. And the jackknife or GJ are resampling methodologies, that usually give a positive answer to such a question.
4.2. Resampling methodologies and corrected-bias EI-estimation. The use of resampling methodologies has revealed to be promising in the esti- mation of the tuning parameter k, and in the reduction of bias of any estimator of a parameter of extreme events. Regarding bias, and due to the fact that at optimal levels, in the sense of minimal mean square error (MSE) we still have a non-null asymptotic bias, we are lead to use the GJ methodology. It is then enough to consider an adequate set of estimators of the parameter of extreme events under consideration, and to build a reduced-bias a ne combination of them. In Gomes et al. (2000, 2002, 2013), also among others, we can find an application of this technique to the EVI-estimation. To illustrate here the use
En✓ˆN(k)o=1+✓ 1 k◆(1+o(1)). n2kn
n2n2k