Zugl.: Karlsruhe, Univ., Diss., 2009 / Hergestellt on demand

Universiẗat, Diss., 2003--Bonn.

Universiẗat, Diss., 1998--Heidelberg.

Techn. Universiẗat, Diss., 2002--Darmstadt.

Universiẗat, Diss., 2003--Stuttgart.

Zugl.: Darmstadt, Techn. Hochsch., Diss., 2006

Universiẗat, Diss., 2004--Bonn.

Universiẗat, Diss., 2004--Osnabrück.

This cumulative dissertation is organized as follows: After the introduction, the second chapter, based on “Asymptotic independence of bivariate order statistics” (2017) by Falk and Wisheckel, is an investigation of the asymptotic dependence behavior of the components of bivariate order statistics. We find that the two components of the order statistics become asymptotically independent for certain combinations of (sequences of) indices that are selected, and it turns out that no further assumptions on the dependence of the two components in the underlying sample are necessary. To establish this, an explicit representation of the conditional distribution of bivariate order statistics is derived. Chapter 3 is from “Conditional tail independence in archimedean copula models” (2019) by Falk, Padoan and Wisheckel and deals with the conditional distribution of an Archimedean copula, conditioned on one of its components. We show that its tails are independent under minor conditions on the generator function, even if the unconditional tails were dependent. The theoretical findings are underlined by a simulation study and can be generalized to Archimax copulas. “Generalized pareto copulas: A key to multivariate extremes” (2019) by Falk, Padoan and Wisheckel lead to Chapter 4 where we introduce a nonparametric approach to estimate the probability that a random vector exceeds a fixed threshold if it follows a Generalized Pareto copula. To this end, some theory underlying the concept of Generalized Pareto distributions is presented first, the estimation procedure is tested using a simulation and finally applied to a dataset of air pollution parameters in Milan, Italy, from 2002 until 2017. The fifth chapter collects some additional results on derivatives of D-norms, in particular a condition for the existence of directional derivatives, and multivariate spacings, specifically an explicit formula for the second-to-last bivariate spacing. / Diese kumulative Dissertation ist wie folgt aufgebaut: Nach der Einleitung wird im zweiten Kapitel, welches auf “Asymptotic independence of bivariate order statistics” (2017) von Falk und Wisheckel beruht, die asymptotische Abhängigkeitsstruktur von bivariaten Ordnungsstatistiken untersucht. Dazu wird eine explizite Darstellung der bedingten Verteilung einer bivariaten Ordnungsstatistik hergeleitet. Kapitel 3, basierend auf “Conditional tail independence in archimedean copula models” (2019) von Falk, Padoan und Wisheckel, zeigt, dass unter schwachen Anforderungen an den Generator einer Archimedischen Copula die übrigen Komponenten unabhängig werden, wenn man auf eine davon bedingt. Das insbesondere auch dann, wenn die Komponenten ohne die Bedingung abhängig waren. Die theoretischen Erkenntnisse werden anhand von Simulationsergebnissen verdeutlicht. “Generalized pareto copulas: A key to multivariate extremes” (2019) von Falk, Padoan und Wisheckel liefert Kapitel 4. Es wird ein nichtparametrischer Ansatz vorgestellt um die Überschreitungswahrscheinlichkeit eines Zufallsvektors über einen festen, hohen Schwellenwert zu schätzen, wenn dieser einer verallgemeinerten Pareto Copula folgt. Das Verfahren wird in den theoretischen Rahmen eingebettet, anhand einer Simulation validiert und auf Luftverschmutzungsdaten in Mailand, Italien, von 2002 bis 2017 angewendet. Im fünften Kapitel werden einige weitere Ergebnisse gesammelt: es geht um Ableitungen von D-Normen, insbesondere um eine Bedingung, die die Existenz der Richtungsableitungen sicherstellt. Außerdem werden multivariate Spacings thematisiert.

In this article a Novikov engine with fluctuating hot heat bath temperature is presented. Based on this model, the performance measure maximum expected power as well as the corresponding efficiency and entropy production rate is investigated for four different stationary distributions: continuous uniform, normal, triangle, quadratic, and Pareto. It is found that the performance measures increase monotonously with increasing expectation value and increasing standard deviation of the distributions. Additionally, we show that the distribution has only little influence on the performance measures for small standard deviations. For larger values of the standard deviation, the performance measures in the case of the Pareto distribution are significantly different compared to the other distributions. These observations are explained by a comparison of the Taylor expansions in terms of the distributions’ standard deviations. For the considered symmetric distributions, an extension of the well known Curzon–Ahlborn efficiency to a stochastic Novikov engine is given.