Page 13 - Textos de Matemática Vol. 47

NAZARE´ AND ARCH PROCESSES: EXTREMAL INDEX ESTIMATION 3
2. The extremal index
Let {Xn}n 1 be a stationary sequence from an underlying model F, under adequate asymptotic dependence conditions. Let {Yn}n 1 be the associated independent, identically distributed (IID) sequence, from the same underlying model F. Let us further denote
MnX ⌘ Xn:n := max(X1,...,Xn) and MnY ⌘ Yn:n := max(Y1,...,Yn). If there exist sequences of constants {an > 0} and {bn 2 R} such that
lim P ✓ MnY bn  x◆ = G(x), with G non-degenerate, n!1 an
then (Gnedenko, 1943)
G(x)⌘EV⇠(x)=⇢ exp (1+⇠x) 1/⇠ ,1+⇠x>0, if ⇠6=0,
exp( exp( x)), x 2 R, if ⇠ = 0,
is the well-known extreme value (EV) cumulative distribution function (CDF), being ⇠ the extreme value index (EVI).
Let us next think on the stationary sequence {Xn}n 1. If the conditions which enable us to guarantee the existence of an extremal index (EI), denoted by ✓, held (Leadbetter et al., 1983), then
lim P✓MnX bn x◆=EV✓⇠(x), 0✓1. n!1 an
Indeed, we shall exclude the ‘slight pathological’ case ✓ = 0.
Remark 2.1. Note that the EV⇠ CDF is max-stable, and consequently,
EV✓⇠(x)=EV⇠✓x ✓◆, ✓=✓⇠ 1, ✓=✓⇠. ✓ ⇠
To better understand the intuitive meaning of the EI, let us think on the point process of exceedances over high thresholds. For any ⌧ > 0, let un = un(⌧), n 1, be a level such that
F(un)=1 ⌧/n+o(1/n), asn!1, (2.1)
a so-called normalized level, i.e. a level such that n(1 F(un))!⌧>0, or equivalently, a level such that F n (un ) ! exp( ⌧ ), as n ! 1. Note that it is possible to choose such a level for any CDF such that (1 F (x ))/(1 F (x)) ! 1, as x ! 1, which will be assumed throughout. Then, with IA denoting the indicator function of A,
Y Xn
Sn = I{Yj>un} j=1