Page 19 - Textos de Matemática Vol. 47

NAZARE´ AND ARCH PROCESSES: EXTREMAL INDEX ESTIMATION 9
of these methodologies in EVT, we apply the GJ methodology to the afore- mentioned EI-estimator in (4.1), as performed in Gomes et al. (2008).
Since the bias term of the aforementioned classical EI-estimator reveals two main components of di↵erent orders, as can be seen in (4.2), we need to use an a ne combination of three EI-estimators, i.e. an order-2 GJ-statistic. Let X = (X1,...,Xn) be a sample from F, and let Tn = Tn(X,F) be an estimator of a functional ✓(F), or of a parameter ✓. If the bias of our estimator reveals two main terms that we would like to remove, the GJ methodology advises us to deal with three estimators with the same type of bias.
E nT (i) ✓o = d (✓) '(i)(n) + d (✓) '(i)(n), i = 1, 2, 3, n1122
the GJ-statistic (of order 2) is given by
T(1) T(2) T(3) 1 1 1 1
Definition 4.1. Given three estimators of ✓, Tn , Tn and Tn , such that
(3) n 1 1 1
with ||A|| denoting, as usual, the determinant of the matrix A. Straightforwardly, one may state the following result.
Proposition 4.2. Under the validity of (4.3), the statistic TGJ, defined in n
(4.4), is unbiased for the estimation of ✓.
The variance of the statistic TGJ is always larger than the variance of the
original estimators, but the MSE of TGJ is often smaller than that of any of
(i)
the statistics Tn , i = 1, 2, 3.
n n n (1) (2)
TGJ := '(1) '(2) '(3) '1 '1 '1 , (4.4)
'(1) '(2) '(3) '(1) '(2) '(3) 222 222
n
n
Given the information on the bias of the extremal index estimator ✓ˆN(k), n
in (4.1), as stated in (4.2), and with bxc denoting the integer part of x, let us consider, just as in Gomes et al. (2008), the levels k, b kc + 1 and b 2kc + 1, dependent of a tuning parameter , 0 < < 1, and the class of estimators,
( 2 + 1) ✓ˆN (b kc + 1) ⇣✓ˆN b 2kc + 1 + ✓ˆN(k)⌘ ✓ˆ G J ( ) ( k ) : = n n n .
n (1 )2
Among the members of this class, the aforementioned authors have been heuris-
(1) (2) (3)
(4.3)
tically led to the choice = 1/4, and to the EI-estimator,
✓ˆGJ(k) := ✓ˆGJ(1/4)(k). (4.5) nn