Goodness-of-Fit Testing for Copula-Based Models with Application in Atmospheric Science

Every elementary probability course discusses how to construct joint distribution functions of independent random variables but joint distribution functions of dependent random variables are usually omitted. Obviously, the reason is that things are not as simple as in the independent case. In this matter, so-called copulas can be an elegant tool to investigate dependency structures other than independence. A copula is a convenient function which links the marginal distributions of random variables to their joint distribution. The beauty here is that one can use suitable copulas to model any desired dependence structure between any set of random variables without even knowing their marginal distributions. In the end, using copulas for modeling comes down to figuring out which copula is suitable given a set of observations. One way to investigate this is based on goodness-of-fit tests which are specifically designed for copulas. Ultimately, this thesis gives an introduction into the necessary theory of copulas and their goodness-of-fit tests in order to use them to compare popular models for cloud overlap in atmospheric science

Copulas in finance and insurance

Copulas provide a potential useful modeling tool to represent the dependence structure among variables and to generate joint distributions by combining given marginal distributions. Simulations play a relevant role in finance and insurance. They are used to replicate efficient frontiers or extremal values, to price options, to estimate joint risks, and so on. Using copulas, it is easy to construct and simulate from multivariate distributions based on almost any choice of marginals and any type of dependence structure. In this paper we outline recent contributions of statistical modeling using copulas in finance and insurance. We review issues related to the notion of copulas, copula families, copula-based dynamic and static dependence structure, copulas and latent factor models and simulation of copulas. Finally, we outline hot topics in copulas with a special focus on model selection and goodness-of-fit testing

Extreme-Value Copulas

Being the limits of copulas of componentwise maxima in independent random samples, extreme-value copulas can be considered to provide appropriate models for the dependence structure between rare events. Extreme-value copulas not only arise naturally in the domain of extreme-value theory, they can also be a convenient choice to model general positive dependence structures. The aim of this survey is to present the reader with the state-of-the-art in dependence modeling via extreme-value copulas. Both probabilistic and statistical issues are reviewed, in a nonparametric as well as a parametric context.Comment: 20 pages, 3 figures. Minor revision, typos corrected. To appear in F. Durante, W. Haerdle, P. Jaworski, and T. Rychlik (editors) "Workshop on Copula Theory and its Applications", Lecture Notes in Statistics -- Proceedings, Springer 201