Page 17 - Textos de Matemática Vol. 47

NAZARE´ AND ARCH PROCESSES: EXTREMAL INDEX ESTIMATION 7 In Figure 2, we picture a sample path of Vn2, Vn given in (3.1), with = 0.68
and = 0.04. Like this we get for Vn, ↵ = 2 and ✓ = 0.84.
7 6 5 4 3 2 1 0
Figure 2. Sample path of the square of an ARCH sequence, V 2 =
Wn2 Vn2 1 + , =0.68, =0.04 (↵=2, ✓=0.84).
n
We further mention that the finite joint structure of the extremes of an ARCH process enabled Gomes et al. (2004) to identify a peculiar ‘unexpected’ phenomenon, in the sense that the EI, one of the most relevant parameters of extreme events, seems not to be the adequate object to consider, when we need to infer on failure probabilities during a finite future time interval (for further details see Gomes et al., 2004, and Gomes et al., 2006, for a short correction).
4. Extremal index estimation
4.1. Classical EI-estimators. Given a sample (X1, . . . , Xn) and chosen a suitable threshold u, a possible estimator of ✓ (Leadbetter and Nandagopalan, 1989) is given by
n 1 I n 1 I PP
j=1 [Xj >u,Xj+1u] j=1 [Xj u<Xj+1]
ˆN ˆN
✓n=✓n(u):= Pn = Pn
.
j=1
I[Xj >u]
j=1
I[Xj >u]
To have consistency, the high level u must be such that n(1 F(un)) = cn⌧ = ⌧n, ⌧n ! 1 and ⌧n/n ! 0 (Nandagopalan, 1990). But whenever dealing with EVI-estimation we often consider an intermediate sequence kn, i.e. a sequence k=kn suchthatkn !1,butk/n!0,asn!1.Suchakn-sequencehas been replaced, in an EI-estimation, by the sequence ⌧n = cn⌧ with cn ! 1 as n ! 1. To make the semi-parametric EI-estimation closer to the most common semi-parametric EVI-estimation, and denoting by {Xi:n}1in the sample of