UC Santa Cruz

Extreme value theory studies the tail behavior of a stochastic process, and plays a key role in a wide range of applications. Understanding and quantifying the behavior of rare events and the associated uncertainties is practically important for risk assessment, since such unexpected events can result in massive losses of wealth and high cost in human life. In this dissertation, we present a Bayesian nonparametric mixture modeling framework for the analysis of extremes with applications in financial industry and environmental sciences. In particular, the modeling is built from the point process approach to analysis of extremes, under which the pairwise observations, comprising the time of excesses and the exceedances over a high threshold, are assumed to arise from a non-homogeneous Poisson process. To relax the time homogeneity restriction, implicit in traditional parametric methods, a nonparametric Dirichlet process mixture model is presented to provide flexibility in estimation of the joint intensity of extremes, the marginal intensity over time, and different types of return level curves for one financial market. This class of models is then expanded to assess the effect of systemic risk in multiple financial markets. In this case, the process generating the extremes is modeled as a superposition of two Poisson process. This approach provides a decomposition of the risk associated with each individual market into two components: a systemic risk component and an idiosyncratic risk component. Finally, we extend the point process framework to model spatio-temporal extremes from environmental processes observed at multiple spatial locations over a certain time interval. Specifically, a spatially varying mixing distribution, assigned a spatial Dirichlet process prior, is incorporated into the model to develop inference for spatial interpolation of risk assessment quantities for high-level exceedances. The modeling approaches are illustrated with a number of simulated and real data examples.