This thesis deals with the tail dependence function. We will prove the consistency of a nonparametric estimator in d-dimensions, do simulations for various applications of the tail dependence function and therefore develop the R package tailDepFunc. After a short introduction, we will present some mathematical background. We introduce copulas and some of their properties. Further, elliptical distributions and elliptical copulas will be discussed as well as Gaussian random fields, which we need for our limit assertions. Different bootstrap methods will be presented as a basis for the multiplier bootstrap method. We define the tail dependence function as a scaled version of tail copulas as introduced in Schmidt and Stadtmüller [2006]. We discuss a non-parametric estimator for the upper tail dependence function under the assumption of known marginals and of unknown marginals. Further, we discuss limit assertions related to the 2-dimensional case in Bücher and Dette [2013] and Schmidt and Stadtmüller [2006] for d dimensions. The tail dependence function for elliptical copulas will be introduced in the fifth part of this chapter. We introduce all properties for the lower dependent case as well. We review some theory regarding multiplier bootstrap empirical processes related to the work of Kosorok [2008]. Following Bücher and Dette [2013], we use this theory to estimate the limit behaviour of the estimators of the tail dependence function. We present some final sample results in the bivariate case for different copula models. We discuss the minimum distance estimator for the family of 1-parametric tail dependence functions as introduced by Bücher and Dette [2013]. With that estimator and its bootstrapped distribution function, we derive analogously to Bücher and Dette [2013] a goodness-of-fit test (GOF-test) for the family of 1-parametric tail dependence functions. We discuss the possibility of applying this theory on tail dependence functions of elliptical copulas. We did some simulations to check the validity of the theory. Finally, we present the R package tailDepFunc, which was written during this thesis. We used it for our simulation. We discuss data simulation and the functions of tailDepFunc. Finally, some larger examples corresponding to some simulations in this thesis are given. At the end, we give a short summary and discussion of our results.     «

This thesis deals with the tail dependence function. We will prove the consistency of a nonparametric estimator in d-dimensions, do simulations for various applications of the tail dependence function and therefore develop the R package tailDepFunc. After a short introduction, we will present some mathematical background. We introduce copulas and some of their properties. Further, elliptical distributions and elliptical copulas will be discussed as well as Gaussian random fields, which we need...     »