Measuring the degree of association between random variables is a task inherent in many practical applications such as risk management and financial modeling. Well-known measures like Spearman's rho and Kendall's tau can be expressed in terms of the underlying copula only, hence, being independent of the underlying univariate marginal distributions. Opposed to these classical measures of association, mutual information, which is derived from information theory, constitutes a fundamentally different approach of measuring association. Although this measure is likewise independent of the univariate margins, it is not a functional of the copula but of the corresponding copula density. Besides the theoretical properties of mutual information as a measure of multivariate association, possibilities to estimate the copula density based on observations of continuous distributions are investigated. To cope with the effect of boundary bias, new estimators are introduced and existing functionals are generalized to the multivariate case. The performance of these estimators is evaluated in comparison to common kernel density estimation schemes. To facilitate variance estimation by means of resampling methods like bootstrapping, an algorithm is introduced, which significantly reduces computation time in comparison with pre-implemented algorithms. In practical applications, complete continuous data is oftentimes not available to the analyst. Instead, categorial data derived from the underlying continuous distribution may be given. Hence, estimation of the copula and its density based on contingency tables is investigated. The newly developed estimators are employed to derive estimates of Spearman's rho and Kendall's tau and their performance is compared.

Putting a particular emphasis on nonparametric methods that rely on modern empirical process techniques, the author contributes to the theory of static and time-varying stochastic models for multivariate association based on the concept of copulas. These functions enable a profound understanding of multivariate association, which is pivotal for judging whether a large set of risky assets entails diversification effects or aggravates risk from an entrepreneurial point of view. Since serial dependence is a stylized fact of financial time series, an asymptotic theory for estimating the structure of association in this context is developed under weak assumptions. A new measure of multivariate association, based on a notion of distance to stochastic independence, is introduced. Asymptotic results as well as hypothesis tests are established which are directly applicable to important types of multivariate financial time series. To ensure that risk management properly captures the current structure of association, it is crucial to assess the constancy of the structure. Therefore, nonparametric tests for a constant copula with either a specified or unspecified change point (candidate) are derived. The thesis concludes with a study of characterizations of association between non-continuous random variables.

Copulas are mathematical objects that fully capture the dependence structure among random variables and hence offer great flexibility in building multivariate stochastic models. Since their introduction in the early 50's, copulas have gained considerable popularity in several fields of applied mathematics, such as finance, insurance and reliability theory. Today, they represent a well-recognized tool for market and credit models, aggregation of risks, portfolio selection, etc. This book is divided into two main parts: Part I - "Surveys" contains 11 chapters that provide an up-to-date account of essential aspects of copula models. Part II - "Contributions" collects the extended versions of 6 talks selected from papers presented at the workshop in Warsaw.

Safety and Reliability of Complex Engineered Systems contains the Proceedings of the 25th European Safety and Reliability Conference, ESREL 2015, held 7-10 September 2015 in Zurich, Switzerland. It includes about 570 papers accepted for presentation at the conference. These contributions focus on theories and methods in the area of risk, safety and

This book presents a novel approach to time series econometrics, which studies the behavior of nonlinear stochastic processes. This approach allows for an arbitrary dependence structure in the increments and provides a generalization with respect to the standard linear independent increments assumption of classical time series models. The book offers a solution to the problem of a general semiparametric approach, which is given by a concept called C-convolution (convolution of dependent variables), and the corresponding theory of convolution-based copulas. Intended for econometrics and statistics scholars with a special interest in time series analysis and copula functions (or other nonparametric approaches), the book is also useful for doctoral students with a basic knowledge of copula functions wanting to learn about the latest research developments in the field.

This book introduces the main theoretical findings related to copulas and shows how statistical modeling of multivariate continuous distributions using copulas can be carried out in the R statistical environment with the package copula (among others). Copulas are multivariate distribution functions with standard uniform univariate margins. They are increasingly applied to modeling dependence among random variables in fields such as risk management, actuarial science, insurance, finance, engineering, hydrology, climatology, and meteorology, to name a few. In the spirit of the Use R! series, each chapter combines key theoretical definitions or results with illustrations in R. Aimed at statisticians, actuaries, risk managers, engineers and environmental scientists wanting to learn about the theory and practice of copula modeling using R without an overwhelming amount of mathematics, the book can also be used for teaching a course on copula modeling.

This book gives readers the solid and formal mathematical background to apply copulas to a range of mathematical areas, such as probability, real analysis, measure theory, and algebraic structures. The authors prove the results as simply as possible and unify various methods scattered throughout the literature in common frameworks, including shuffles of copulas. They also explore connections with related functions, such as quasi-copulas, semi-copulas, and triangular norms, that have been used in different domains.

Copula Methods in Finance is the first book to address the mathematics of copula functions illustrated with finance applications. It explains copulas by means of applications to major topics in derivative pricing and credit risk analysis. Examples include pricing of the main exotic derivatives (barrier, basket, rainbow options) as well as risk management issues. Particular focus is given to the pricing of asset-backed securities and basket credit derivative products and the evaluation of counterparty risk in derivative transactions.