Extremal Behavior of Time Series: Refined Models, Analysis and Inference

该项目的目的是发展新的概率和统计方法，这些方法允许对渐近独立时间序列的极值行为进行精细描述，即在极限中没有极值聚类的时间序列。到目前为止，一个很好的关于时间序列极值的理论只存在于渐近相关的情况下。然而，极值理论的“经典”渐近方法忽略了一个事实，即在时间序列的“前渐近”行为中可能演变出各种不同的行为。例如，许多渐近独立的时间序列模型在有限的样本量中显示出相当数量的大值聚类。通过结合渐近独立随机向量的模型和渐近相关时间序列的方法，我们旨在改进极值理论的工具，以便更好地描述和区分不同类型的渐近独立性。我们通过发展经典理论只给出退化结果的新极限过程来解决概率方面的问题，并通过给出极值依赖性不同方面的估计和模型估计和验证的统计过程来解决统计方面的问题。渐近独立时间序列的一个突出例子是众所周知的随机波动模型。因此，特别关注金融时间序列的分析，特别是关于这门课对金融数据极端建模的适用性。

The aim of this project is the development of new probabilistic and statistical methods which allow for a refined description of the extremal behavior of asymptotically independent time series, i.e. time series which show no clustering of extreme values in the limit. So far, a well explored theory about time series extremes exists only in the asymptotically dependent case. However, the “classical” asymptotic approach of extreme value theory neglects the fact that a variety of different behaviors may evolve in the “pre-asymptotic” behavior of a time series. For example, many asymptotically independent time series models show a decent amount of clustering of large values in finite samples sizes. By a combination of models for asymptotically independent random vectors and methods of asymptotically dependent time series, we aim at a refinement of the tools of extreme value theory that allow for a better description and distinction of different types of asymptotic independence. We tackle both the probabilistic aspects by the development of new limit processes for cases in which classic theory gives only degenerate results, and the statistical side by giving estimators for different aspects of extremal dependence and statistical procedures for model estimation and validation. A prominent example of asymptotically independent time series is the well-known class of stochastic volatility models. Therefore, a special focus is laid on the analysis of financial time series, especially with regard to the suitability of this class for modeling the extremes of financial data.