TAIL DENSITIES OF COPULAS AND THEIR APPLICATIONS TO EXTREMAL DEPENDENCE ANALYSIS OF VINES
Extreme events occur everywhere, and analyzing impacts of these events becomes fundamentally important in many areas such as financial risk management, electrical grid reliability analysis, and environmental impact assessment. The risk factors in such fields exhibit strong dependence which often lies among their extreme values. Modeling dependence of extreme events is therefore critically to avoid or mitigate long-term, contagious damages. This research targets this fundamental topic about analyzing tail risk and introduces a new copula method based on tail density to study extremal dependence.Copulas become an efficient tool to study dependence of multivariate distributions. For higher dimensional data, however, standard copula-based methods are cumbersome and time consuming, thus vine copulas are often used. In this dissertation we introduce a copula tail density approach to analyze extremal dependence of the copulas that are specified only by densities. The idea is based on tail density of multivariate regular variation with standardized margins that extracts scale-invariant tail dependence. The main contributions of this dissertation are as follows.1. Coupled with regularly varying margins, the relation between the copula tail density and the tail density of multivariate regular variation is established. The advantage of the copula tail density lies in its ability to separate scale-invariant tail dependence from marginal distributions.2. We derive tractable formulas of tail densities of Archimedean and t copulas explicitly. 3. We apply our tail density approach to vine copulas, and obtain the tail density of D-vines in the recursive forms according to their underlying graph structures.4. We apply our tail density method to analyze Value-at-Risk (VaR) of aggregated loss of portfolios with multivariate regularly varying loss distributions. The explicit expressions of tail approximations of VaR for a loss vector are obtained in terms of tail densities.In risk analysis, remote critical sets are usually neither upper nor lower orthant sets, and estimating tail risk in such situations using standard extreme value methods becomes problematic. The significance of our results is that we provide an effective tail density based copula method to estimate and analyze tail risk for high dimensional stochastic systems that are modeled via vine copulas.