Page 63 - Textos de Matemática Vol. 47

BOOTSTRAP AND JACKKNIFE METHODS IN EXTREMAL INDEX ESTIMATION 53 Let {Zi}i 1 be a sequence of independent, unit-Fr´echet distributed random
variables. For 0 < ✓  1, let
X1 =Z1 Xi =max{(1 ✓)Xi 1,✓Zi}, i 2.
Forun =nx,0<x<1,Pr Mn un !exp ✓/x ,asn!1,sothe extremal index of the sequence is ✓.
iid 2dep
0 10 20 30 40 k
Figure 1. One realization of an i.i.d. process and a 2-dependent process.
17 1.072552 0.686968 3.097931 18 0.349065 1.948675 2.788138
19 0.496704 0.974338 2.509324 20 20 1.823144 0.779399 2.258392 15 21 0.817736 0.964604 6.40181
22 0.68322 1.447588 5.761629 10 23 0.453179 1.237074 5.185466 5 24 5.197649 0.618537 4.66692 25 4.337472 0.369991 4.200228
=0.9
0 26 0.614502 0.560788 3.780205
0 10 20 30 40 50
27 0.692333 0.684494 3.402184
20 15 10
5
=0.5
0
0 10 20 30 40 50
=0.1 20
15 10 5 0
0 10 20 30 40 50
Figure 2. One realization of an ARMAX process with ✓ = 0.9; 0.5 and 0.1.
Figure 2 shows a partial realization of variables following an ARMAX process with ✓ = 0.9; 0.5 and 0.1, respectively. The maxima show increasing clustering as ✓ ! 0. Notice also again a ‘shrinkage of maximum values’ as dependence increases.
One of the local dependence conditions guaranteeing the existence of an extremal index is the D00 condition, introduced by Leadbetter and Nandagopalan (1989), under which it can be given a very common interpre- tation of ✓, as being the reciprocal of the ‘mean time of duration of extreme
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