Abstract
The main topic of this thesis is quantification of strength of dependence within a d-variate random vector X. The dependence structure of X is fully determined by the corresponding copula function.
We aim to discuss ways to summarize the strength of dependence from a copula into a single number. In particular, we focus on two types of such coefficients.
The overall dependence is described by multivariate association measures which quantify the tendency of the components of X to simultaneously take large or small values. The other considered type of coefficients are multivariate tail coefficients, which only focus on tail, i.e. extremal, behavior.
In other words, they quantify the tendency of the components of X to simultaneously take extremely large or extremely small values. For both of these types of coefficients, we first study their theoretical properties, in particular the effect of adding other random variables to X.
For Archimedean and meta-elliptical copulas, we also focus on the asymptotic behavior when the dimension of X grows to infinity. Further, three novel multivariate tail coefficients are proposed in the thesis.
We introduce a nonparametric estimator for Gini's gamma, as well as for several multivariate tail coefficients. Asymptotic properties, such as consistency or asymptotic normality, are proven for these estimators.
A special attention is given to one of the multivariate tail coefficients called extremal dependence coefficient. For this coefficient, being a limit of conditional probabilities, we give practical procedures to to estimate this limit, aiming at minimization of asymptotic mean square error of the estimator.
Finally, we apply extremal dependence coefficient in a clustering algorithm, grouping components of X by means of their tail behavior.