Extremal dependence of multivariate distributions and its applications

Stochastic dependence arises in many fields including electrical grid reliability, network/internet traffic analysis, environmental and agricultural impact assessment, and financial risk management. Models in these fields exhibit the stochastic behavior of strong dependence often among their extreme values. Extremal dependence is directly related to the dynamics of a system. It is critically important to understand this relationship. If not, the extremal dependence can cause long term, contagious damage. The tail dependence of multivariate distributions is frequently studied via the tool of copulas, but copula-based methods become inefficient in higher dimensions. The theories of multivariate regular variation and extreme values bring in many results for multivariate distributions based on various measures. Applying these theories, however, is difficult because the measures are not explicit. In this dissertation, we establish the link between tail dependence parameters and the theory of multivariate regular variation. For a multivariate regularly varying distribution, we show that the upper tail dependence function and the intensity measure are equivalent in describing its extremal dependence structure. With this new characterization we can efficiently evaluate tail dependence for multivariate distributions when the copulas are not explicitly accessible. As applications, we derive tractable formulas for tail dependence parameters of heavy-tailed scale mixtures of multivariate distributions. We discuss the structural traits of multivariate tail dependence, such as the monotonicity properties of tail dependence for bivariate elliptical distributions. We also give a general method to approximate the Value-at-Risk for aggregated data in terms of the tail dependence function of the data and the Value-at-Risk for one of the variables. Explicit approximations are obtained for Archimedean copulas, Archimedean survival copulas and Pareto distributions of Marshall-Olkin type. Additionally, to visualize the monotonicity properties of the tail dependence parameters, we give some simulation results for Pareto distributions in three and four dimensions