Structures of Multivariate Dependence

The investigation of dependence structures plays a major role in contemporary statistics. During the last decades, numerous dependence measures for both univariate and multivariate random variables have been established. In this thesis, we study the distance correlation coefficient, a novel measure of dependence for random vectors of arbitrary dimension, which has been introduced by Szekely, Rizzo and Bakirov and Szekely and Rizzo. In particular, we define an affinely invariant version of distance correlation and calculate this coefficient for numerous distributions: for the bivariate and the multivariate normal distribution, for the multivariate Laplace and for certain bivariate gamma and Poisson distributions. Moreover, we present a useful series representation of distance covariance for the class of Lancaster distributions and derive a generalization of an integral, which plays a fundamental role in the theory of distance correlation. We further investigate a variable clustering problem, which arises in low rank Gaussian graphical models. In the case of fixed sample size, we discover that this problem is mathematically equivalent to the subspace clustering problem of data for independent subspaces. In the asymptotic setting, we derive an estimator, which consistently recovers the cluster structure in the case of noisy data