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

Abstract

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.