## FANDOM

32.681 Seiten

 Typus Verschleierung Bearbeiter Graf Isolan Gesichtet
Untersuchte Arbeit:
Seite: 129, Zeilen: 24-35
Quelle: Smith 2003
Seite(n): 50, Zeilen: 15-18, 22-28
We are interested in the extremal properties of this process. Without loss of generality, we consider the case where all marginal distributions have unit Frèchet [sic!] distribution.

Definition 5.2.1. The process Yij is called max-stable if all finite dimensional distributions are max-stable, i.e., for any n ≥ 1, r ≥ 1

${\scriptstyle \mathbb P}$(Yij ≤ ruij: 1≤i≤n, 1≤j≤d)r = ${\scriptstyle \mathbb P}$(Yij ≤ uij: 1≤i≤n, 1≤j≤d).

Furthermore, a process Xij for i = 1,...,n, j = 1,...,d, is said to be in the domain of attraction of a max-stable process Yij if there exist normalising constants anij > 0, bnij such that for any finite r

${\scriptstyle \lim\limits_{n\to\infty}\mathbb{P}\left(\frac{X_{ij}-b_{nij}}{a_{nij}}\le ru_{ij}:1\le i\le n, 1\le j\le j [sic!] \right)^r}$
${\scriptstyle = \mathbb{P}(Y_{ij}\le u_{ij}: 1\le i\le n, 1\le j\le d). \qquad (5.2.1)}$

Since we assume a priori that the process Xij also has unit Frèchet [sic!], then we may take anij = n, bnij = 0.

If we are interested in the extremal properties of this process, then for the same reasons as with multivariate extremes, it suffices to consider the case where all the marginal distributions have been transformed to unit Fréchet. Such a process is called max-stable if all the finite-dimensional distributions are max-stable, i.e. for any n ≥ 1, r ≥ 1

[...]

Pr(Yid ≤ nyid: 1≤i≤r, 1≤d≤D)n = Pr(Yid ≤ yid: 1≤i≤r, 1≤d≤D). (7.7)

Corresponding to this, a process {Xid : i = 1,2,...; 1≤d≤D} is said to be in the domain of attraction of a max-stable process {Yid : i = 1,2,...; 1≤d≤D} if there exist normalising constants anid > 0, bnid such that for any finite r

${\scriptstyle \lim\limits_{n\to\infty}Pr\left\{\frac{X_{id}-b_{nid}}{a_{nid}}\le ny_{id}:1\le i\le r, 1\le d\le D \right\}^n}$
${\scriptstyle = Pr(Y_{id}\le y_{id}: 1\le i\le r, 1\le d\le D). \qquad (7.8)}$

If we assume a priori that the X process also has unit Fréchet margins, then we may take anid = n, bnid = 0.

 Anmerkungen Keine Kennzeichnung der Übernahme, kein Hinweis auf die Quelle. Obwohl es sich hier im Wesentlichen um eine Definition handelt, so sind doch auch erklärende Bemerkungen wörtlich übernommen worden. Fréchet ist in der Quelle richtig geschrieben; bei der Übernahme ist das Wort "margins" verlorengegangen Sichter (Graf Isolan) Plagiatsfischer