Extreme value theory studies the probabilistic and statistical behaviors of tail observations. The theory is widely applied in various areas such as finance and environmental science. In practice, tail observations are often dependent in time, space or both. Therefore, characterizing the dependence structure of the data plays a major role in the statistical modeling of extremes. In this thesis, we present three studies related to finance and air quality events observed in time and space-time. The primary goals of the studies are to understand the extremal behaviors through the proposed models and provide predictions.

First, we propose a threshold extreme value distribution to study the tail asymmetry between the downside risks of long and short positions for securities markets....[ Read more ]

Extreme value theory studies the probabilistic and statistical behaviors of tail observations. The theory is widely applied in various areas such as finance and environmental science. In practice, tail observations are often dependent in time, space or both. Therefore, characterizing the dependence structure of the data plays a major role in the statistical modeling of extremes. In this thesis, we present three studies related to finance and air quality events observed in time and space-time. The primary goals of the studies are to understand the extremal behaviors through the proposed models and provide predictions.

First, we propose a threshold extreme value distribution to study the tail asymmetry between the downside risks of long and short positions for securities markets. We handle the temporal dependence of the financial returns by standardizing the proposed distribution and integrating it into a generalized autoregressive conditional heteroskedasticity (GARCH) framework. We apply our model across different asset classes to predict value-at-risk with multiple horizons and assess the evidence of tail asymmetry through Bayesian model selection techniques. Financial and statistical implications of the empirical results are discussed.

Second, we address the Bayesian inference for spatial extremes based on composite likelihood under extensive simulations. Our models of interest are the max-stable processes which are the limiting cases of infinite-dimensional extremes. We evaluate various parametric models of max-stable processes with different priors, sample sizes and composite weights.

Finally, we propose a novel copula-based spatial extreme value model that can handle both the spatial dependence and interactions among multiple quantities in a unified way. The proposal addresses the surprisingly low availability of spatial extreme model in the literature that can capture both types of dependence. We apply our model to analyze weekly maxima of five types of air pollutants recorded in Pearl River Delta in China and predict the levels of pollution for unmonitored regions.