@phdthesis{Aulbach2015,
  author    = {Aulbach, Stefan},
  title     = {Contributions to Extreme Value Theory in Finite and Infinite Dimensions: With a Focus on Testing for Generalized Pareto Models},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-127162},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2015},
  abstract  = {Extreme value theory aims at modeling extreme but rare events from a probabilistic point of view. It is well-known that so-called generalized Pareto distributions, which are briefly reviewed in Chapter 1, are the only reasonable probability distributions suited for modeling observations above a high threshold, such as waves exceeding the height of a certain dike, earthquakes having at least a certain intensity, and, after applying a simple transformation, share prices falling below some low threshold. However, there are cases for which a generalized Pareto model might fail. Therefore, Chapter 2 derives certain neighborhoods of a generalized Pareto distribution and provides several statistical tests for these neighborhoods, where the cases of observing finite dimensional data and of observing continuous functions on [0,1] are considered. By using a notation based on so-called D-norms it is shown that these tests consistently link both frameworks, the finite dimensional and the functional one. Since the derivation of the asymptotic distributions of the test statistics requires certain technical restrictions, Chapter 3 analyzes these assumptions in more detail. It provides in particular some examples of distributions that satisfy the null hypothesis and of those that do not. Since continuous copula processes are crucial tools for the functional versions of the proposed tests, it is also discussed whether those copula processes actually exist for a given set of data. Moreover, some practical advice is given how to choose the free parameters incorporated in the test statistics. Finally, a simulation study in Chapter 4 compares the in total three different test statistics with another test found in the literature that has a similar null hypothesis. This thesis ends with a short summary of the results and an outlook to further open questions.},
  subject      = {Extremwertstatistik},
  language  = {en}
}
@phdthesis{Zott2016,
  author    = {Zott, Maximilian},
  title     = {Extreme Value Theory in Higher Dimensions - Max-Stable Processes and Multivariate Records},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-136614},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2016},
  abstract  = {Die Extremwerttheorie behandelt die stochastische Modellierung seltener und extremer Ereignisse. W{\"a}hrend fundamentale Theorien in der klassischen Stochastik, wie etwa die Gesetze der großen Zahlen oder der zentrale Grenzwertsatz das asymptotische Verhalten der Summe von Zufallsvariablen untersucht, liegt in der Extremwerttheorie der Fokus auf dem Maximum oder dem Minimum einer Menge von Beobachtungen. Die Grenzverteilung des normierten Stichprobenmaximums unter einer Folge von unabh{\"a}ngigen und identisch verteilten Zufallsvariablen kann durch sogenannte max-stabile Verteilungen charakterisiert werden. In dieser Dissertation werden verschiedene Aspekte der Theorie der max-stabilen Zufallsvektoren und stochastischen Prozesse behandelt. Insbesondere wird der Begriff der 'Differenzierbarkeit in Verteilung' eines max-stabilen Prozesses eingef{\"u}hrt und untersucht. Ferner werden 'verallgemeinerte max-lineare Modelle' eingef{\"u}hrt, um einen bekannten max-stabilen Zufallsvektor durch einen max-stabilen Prozess zu interpolieren. Dar{\"u}ber hinaus wird der Zusammenhang von extremwerttheoretischen Methoden mit der Theorie der multivariaten Rekorde hergestellt. Insbesondere werden sogenannte 'vollst{\"a}ndige' und 'einfache' Rekorde eingef{\"u}hrt, und deren asymptotisches Verhalten untersucht.},
  subject      = {Stochastischer Prozess},
  language  = {en}
}