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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.
Extreme value theory is concerned with the stochastic modeling of rare and extreme events. While fundamental theories of classical stochastics - such as the laws of small numbers or the central limit theorem - are used to investigate the asymptotic behavior of the sum of random variables, extreme value theory focuses on the maximum or minimum of a set of observations. The limit distribution of the normalized sample maximum among a sequence of independent and identically distributed random variables can be characterized by means of so-called max-stable distributions.
This dissertation concerns with different aspects of the theory of max-stable random vectors and stochastic processes. In particular, the concept of 'differentiability in distribution' of a max-stable process is introduced and investigated. Moreover, 'generalized max-linear models' are introduced in order to interpolate a known max-stable random vector by a max-stable process. Further, the connection between extreme value theory and multivariate records is established. In particular, so-called 'complete' and 'simple' records are introduced as well as it is examined their asymptotic behavior.
Die vorliegende Arbeit hat zum Ziel, das Antwortverhalten nichtlinearer Reaktionen auf zielgerichtete Störungen zu untersuchen. Dabei beschäftigt sie sich mit zwei nichtlinearen chemischen Sauerstoff-Oszillatoren. Bei den beiden nichtlinearen chemischen Reaktionen handelt es sich um den Polyacrylamid-Methylenblau-Sauerstoff- (PA-MBO) Oszillator und um die Kupfer(II)ionen katalysierte Oxidation von Ascorbinsäure durch Luftsauerstoff. Im ersten Fall wird durch selektive Belichtung des Reaktionsmediums die gebildete Geloberfläche durch ein computergenerirtes Muster kodiert. Die Systemantwort wird mit Hilfe einer CCD-Kamera aufgenommen und danach einer Analyse unterzogen. Die erhaltenen Ergebnisse werden anschließend durch eine Computersimulation verifiziert. Die zweite untersuchte Möglichkeit, das PA-MBO-System einer Störung zu unterwerfen, ist das Anlegen eines externen elektrischen Feldes. In einer speziell dafür entworfenen Anordnung bildet sich ein quasi-eindimensionales Turing-Muster. In dieser quasi-eindimensionalen Anordnung kann die Reaktion leicht elektrischen Strömen von bis zu 200 mA/cm2 ausgesetzt werden. Die experimentellen Daten werden anschließend der Karhunen-Loeve Zerlegung unterworfen, um die komplexe Dynamik der Systemantwort zu studieren. Die Oxidation von Ascorbinsäure durch Luftsauerstoff in Gegenwart von Kupfer(II)ionen, wird im CSTR durchgeführt. Dabei läßt sich das Phänomen der stochastischen Resonanz beobachten, wenn man die Flußrate sinusförmig moduliert und dieser Frequenz zusätzlich weißes Rauschen überlagert.