Page 60 - Textos de Matemática Vol. 47

50 M. M. NEVES
unknown distribution function (d.f.) F and we are concerned with the lim- iting behaviour of either Mn ⌘ Xn:n = max(X1,...,Xn) or mn ⌘ X1:n = min(X1, . . . Xn) as n ! 1. Dealing with maximum values, whenever it is pos- sible to linearly normalize Mn so that we get a non-degenerate limit, such a limit is the Extreme Value (EV) d.f., given by
EV⇠(x):=⇢ exp[ (1+⇠x) 1/⇠], 1+⇠x>0, if ⇠6=0, (1.1) exp[ exp( x)], x 2 R, if ⇠ = 0.
We then say that F is in the domain of attraction for maxima of EV⇠, denoting this by F 2 DM (EV⇠ ).
The EV⇠ d.f., in (1.1), incorporates the three Fisher-Tippett types: the Gum- bel family, ⇤(x) ⌘ EV0(x) := exp( exp( x)), x 2 R, (⇠ = 0), the limit for exponentially-tailed distributions; the Fr´echet family, ↵(x) ⌘ EV1/↵(↵(x 1)) := exp( x ↵), ↵ > 0, x > 0, (⇠ = 1/↵ > 0), the limit for negative polynomial heavy-tailed distributions and the Weibull family, ↵(x) ⌘ EV 1/↵ (↵(x + 1)) := exp( ( x)↵), ↵ > 0, x < 0, (⇠ = 1/↵ < 0), the limit for short-tailed distributions. If ⇠ = 0, the right endpoint, x⇤ := sup{x : F(x) < 1}, can then be either finite or infinite. If ⇠ > 0, F has an infinite right endpoint. If ⇠ < 0, F has a finite right endpoint. The shape parameter, ⇠, is directly related to the weight of the right tail, F := 1 F , of the underlying model F . As ⇠ increases the right tail becomes heavier.
Whenever independence is no longer valid, some important dependent se- quences have been studied. The limit distributions of their order statistics, under some dependence structures, have been obtained. Stationary sequences are examples of such sequences and are realistic for many real problems.
As dependence in stationary sequences can assume several forms, some con- ditions have to be imposed. The first condition, known as the D(un) dependence condition, Leadbetter et al. (1983), ensures that any two extreme events can become approximately independent as n increases when separated by a rela- tively wide interval of length ln = o(n). Hence, D(un) limits the long-range dependence between such events.
Under adequate local dependence conditions, the limiting d.f. of the max- imum of the stationary sequence may be directly related to the maximum of the i.i.d. associated sequence, through a new parameter, the so-called extremal index, usually denoted by ✓. This parameter also a↵ects other parameters of extreme events, so its reliable estimation is of great importance.
However, estimators of ✓ show the usual behaviour of semi-parametric tail estimators: nice asymptotic properties, but a high variance for small k, the