Page 65 - Textos de Matemática Vol. 47

BOOTSTRAP AND JACKKNIFE METHODS IN EXTREMAL INDEX ESTIMATION 55
In this paper our attention will be focused on the UC-estimator in (2.3). Given the sample (X1,...,Xn) and the associated ascending order statistics, X1:n  · · ·  Xn:n , we shall consider the deterministic level u ⌘ un substituted by the stochastic one, Xn k:n, and write the UC estimator, in (2.3), as a function of k (Gomes et al., 2008),
b U C b U C 1 nX 1
⇥ ⌘ ⇥ (k):= k I(Xi Xn k:n <Xi+1). (2.4)
i=1
For many dependent structures, the bias of ⇥bUC(k) has two dominant com-
(2.5)
ponents of orders k/n and 1/k (see Gomes et al., 2008), Bias[⇥bUC(k)]='1(✓)k +'2(✓)1 +o✓k◆+o✓1◆,
nknk whenever n ! 1 and k ⌘ k(n) ! 1, k = o(n).
Bias2
2 kkk
Figure 4. MSE, Var and Bias2 of ⇥bUC(k) for ✓ = 0.9,0.5,0.1 for the ARMAX process in Example 2.
Figure 3. Mean values of ⇥bUC(k) for ✓ = 0.9,0.5,0.1 for the ARMAX process in Example 2.
MSE
Var Var
Bias
MSE
Bias2
MSE
Var