TY  - JOUR
A1  - Falk, Michael
A1  - Marohn, Frank
T1  - Von Mises condition revisited
N2  - It is shown that the rate of convergence in the von Mises conditions of extreme value theory determines the distance of the underlying distribution function F from a generalized Pareto distribution. The distance is measured in terms of the pertaining densities with the limit being ultimately attained if and only if F is ultimately a generalized Pareto distribution. Consequently, the rate of convergence of the extremes in an lid sample, whether in terms of the distribution of the largest order statistics or of corresponding empirical truncated point processes, is determined by the rate of convergence in the von Mises condition. We prove that the converse is also true.
KW  - Von Mises conditions
KW  - extreme value theory
KW  - extreme value distribution
KW  - extreme order statistics
KW  - generalized Pareto distribution
Y1  - 1993
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-45790
ER  - 
TY  - THES
A1  - Tichy, Diana
T1  - On the Fragility Index
T1  - Über den Fragilitätsindex
N2  - The Fragility Index captures the amount of risk in a stochastic system of arbitrary dimension. Its main mathematical tool is the asymptotic distribution of exceedance counts within the system which can be derived by use of multivariate extreme value theory. Thereby the basic assumption is that data comes from a distribution which lies in the domain of attraction of a multivariate extreme value distribution. The Fragility Index itself and its extension can serve as a quantitative measure for tail dependence in arbitrary dimensions. It is linked to the well known extremal index for stochastic processes as well the extremal coefficient of an extreme value distribution.
N2  - Der Fragilitätsindex erfasst das Risiko des Zusammenbruchs eines stochastischen Systems beliebiger Dimension. Wesentlicher Baustein dieser mathematischen Größe ist dabei die asymptotische Verteilung der Überschreitungsanzahl innerhalb des stochastischen Systems. Die Herleitung basiert auf wesentlichen Erkenntnissen aus der multivariaten Extremwerttheorie. Die Hauptannahme besteht darin, dass Realisationen einer Zufallsgröße von einer Verteilung erzeugt werden, welche im Anziehungsbereich einer multivariaten Extremwertverteilung liegt. Der Fragilitätsindex und seine Erweiterung stellen ein quantitatives Maß beliebiger Dimension für Abhängigkeiten zwischen extremen Ereignissen dar. Er steht dabei in direkter Verbindung zum Extremalindex für stochastische Prozesse und zum Extremalkoeffizienten für Extremwertverteilungen.
KW  - Extremwertstatistik
KW  - Stochastisches System
KW  - Fragilitätsindex
KW  - Extremwerttheorie
KW  - Abhängigkeitsmaß
KW  - Überschreitungsanzahl
KW  - Fragility Index
KW  - tail dependence
KW  - extreme value theory
KW  - exceedance counts
KW  - extremal coefficient
Y1  - 2011
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-73610
ER  - 
TY  - THES
A1  - Hofmann, Martin
T1  - Contributions to Extreme Value Theory in the Space C[0,1]
T1  - Beiträge zur Extremwerttheorie im Raum C[0,1]
N2  - We introduce some mathematical framework for extreme value theory in the space of continuous functions on compact intervals and provide basic definitions and tools. Continuous max-stable processes on [0,1] are characterized by their “distribution functions” G which can be represented via a norm on function space, called D-norm. The high conformity of this setup with the multivariate case leads to the introduction of a functional domain of attraction approach for stochastic processes, which is more general than the usual one based on weak convergence. We also introduce the concept of “sojourn time transformation” and compare several types of convergence on function space. Again in complete accordance with the uni- or multivariate case it is now possible to get functional generalized Pareto distributions (GPD) W via W = 1 + log(G) in the upper tail. In particular, this enables us to derive characterizations of the functional domain of attraction condition for copula processes. Moreover, we investigate the sojourn time above a high threshold of a continuous stochastic process. It turns out that the limit, as the threshold increases, of the expected sojourn time given that it is positive, exists if the copula process corresponding to Y is in the functional domain of attraction of a max-stable process. If the process is in a certain neighborhood of a generalized Pareto process, then we can replace the constant threshold by a general threshold function and we can compute the asymptotic sojourn time distribution.
N2  - Es wird ein Zugang zur Extremwerttheorie auf dem Raum C[0,1] gegeben. Nach Charakterisierung und Analyse standard max-stabiler Prozesse, wird ein "funktionaler Anziehungsbereich" für standard max-stabile Prozesse vorgeschlagen, der allgemeiner ist als der übliche, der mittels schwacher Konvergenz definiert wird. Schließlich werden Verweildauern stetiger Prozesse über hohen Schwellenwerten betrachtet.
KW  - Extremwertstatistik
KW  - stochastischer Prozess
KW  - Extremwerttheorie
KW  - extreme value theory
KW  - stochastic processes
KW  - functional D-norm
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-74405
ER  -