Page 64 - Textos de Matemática Vol. 47

54 M. M. NEVES
events’ which is directly related to the exceedances of high levels (Hsing et al., 1988, Leadbetter and Nandagopalan, 1989),
✓=1. limiting mean size of clusters
Identifying clusters by the occurrence of downcrossings or upcrossings, we can write
✓= limPr[X2 un|X1 >un]= limPr[X1 un|X2 >un]. (2.2) n!1 n!1
Given a sample, (X1, . . . , Xn), the empirical counterpart of the above inter- pretation led to the naive estimators: the classical up-crossing (down-crossing)
estimator, UC estimator , ⇥bUC (DC-estimator, ⇥bDC) (Nandagopalan, 1990; Gomes, 1990, 1992, 1993)
n 1
P I(Xi un <Xi+1)
⇥bUC(un) := i=1
I(Xi > un)
for a suitable threshold un, where I(A) denotes, as usual, the indicator function of A. Consistency of this estimator is obtained provided that the high level un is a normalized level, i.e. if with ⌧ ⌘ ⌧n > 0, the underlying d.f. F verifies
F(un)=1 ⌧/n+o(1/n), n!1 and ⌧/n!0.
Di↵erent forms of identifying clusters gave rise to other estimators. Let us men- tion two very popular estimators: the blocks estimator and the runs estimator, Hsing (1991, 1993). The blocks estimator is derived by dividing the data set into approximately kn blocks of length rn, where n ⇡ kn ⇥ rn, i.e., considering kn = [n/rn]. Each block is treated as one cluster and the number of blocks in which there is at least one exceedance of the threshold un is counted. The blocks estimator, ⇥bBn (un), is then defined as
Pkn I max X ,···,X >u ⇥bB(u):= i=1 P(i 1)rn+1 irn n.
n n ni=1 I (Xi > un)
If we assume that a cluster consists of a run of observations between two
exceedances, then the runs estimator is defined as:
Pn i=1
n 1
P I(Xi >un,Xi+1 un)
⌘ i=1 := ⇥bDC(un)
I(Xi > un)
(2.3)
⇥n(un):= Pni=1I(Xi >un) .
Pn i=1
bR Pni=1 I Xi > un,max Xi+1,··· ,Xi+rn 1  un
Under mild conditions, limn!1 ⇥bBn (un) = limn!1 ⇥bRn (un) = ✓. Other proper- ties of these estimators have been well studied by Smith and Weissman (1994) and Weissman and Novak (1998).