Inference Problems in Extreme Value Statistics

极值理论的研究近年来备受关注。估计罕见事件的概率是主要兴趣之一。这促使许多研究人员开发极值统计的新方法。应用极值理论的困难之一是必须仔细选择样本分数，以使估计具有尽可能快的收敛速度，同时其偏差可以忽略不计。这项建议包括五个主题，如下所示。首先，研究人员提出了一种数据倾斜方法，当基础分布属于其中一个极值分布的吸引域时，构造极值尾概率的置信度区间。所提出的方法有望在覆盖概率方面产生更准确的置信度区间，并且对样本分数的选择具有更强的鲁棒性。其次，研究人员提出了估计二元极值相依结构的新方法。在二元极值统计中，谱度量等相依结构的估计是一个重要问题。谱测量与两个边际极限一起确定了双变量极值的极限分布。第三，研究人员提出了依赖函数的一阶偏导数的光滑估计，以便基于正态近似构造依赖函数的置信度区间。第四，研究了如何构造二元极值统计量的谱测度和尾部相关函数的置信带。采用特殊的自举技术来解决这些问题。这使得人们可以在不估计全局依赖函数的导数的情况下获得渐近正确的置信带。第五，研究人员提出了边缘分布未知的高维极值分布的Pickands相关函数的新估计量，极值统计在气象、水文学、气候学、环境科学、电信、保险、金融等领域都有应用。研究人员为单变量和多变量极值统计中的风险分析开发了更准确和更有效的方法。该提案中项目的进展加强了研究人员和研究人员在这些领域的合作。拟议的活动还包括教授研究生在他们未来的研究中使用极值统计。这项提案中开发的新方法预计也将有更广泛的应用。例如，精算师可以使用它们来计算和防范罕见但在财务上具有破坏性的事件的概率，或者统计学家可以使用它们来计算防止洪水泛滥所需的海堤高度。它们还可以用来告诉工程师建造桥梁或石油钻井平台的强度，以及模拟过高的污染水平。

The study of extreme-value theory has been paid much attention in recent years. Estimating probabilities of rare events is one of the primary interests. This has motivated many researchers to develop new methodologies in extreme-value statistics. One of the difficulties in applying the extreme-value theory is that the sample fraction has to be carefully chosen such that the estimation has a convergence rate as fast as possible while its bias is negligible. This proposal consists of five topics, as follows. First, the investigator proposes a data tilting method to construct confidence intervals for extreme tail probabilities when the underlying distribution belongs to the domain of attraction for one of the extreme-value distributions. The proposed method is expected to generate more accurate confidence intervals in terms of coverage probabilities and to be more robust against the choice of the sample fraction. Second, the investigator develops new methods for estimation of dependence structures in bivariate extremes. Estimation of the dependence structures, such as the spectral measure, in bivariate extreme-value statistics is an important issue. The spectral measure, together with the two marginal limits, determines the limiting distribution of the bivariate extremes. Third, the investigator proposes smooth estimators for the first partial derivatives of the dependence function in order to construct confidence intervals for the dependence function based on the normal approximation. Fourth, the investigator studies how to construct confidence bands for the spectral measure and tail dependence functions in bivariate extreme-value statistics. Special bootstrap techniques are applied to solve the problems. This allows one to obtain asymptotically correct confidence bands without estimating the derivatives of the dependence function globally. Fifth, the investigator proposes new estimators for the Pickands dependence function of high dimensional extreme-value distributions with unknown marginal distributions.Extreme-value statistics have found applications in many fields such as meteorology, hydrology, climatology, environmental sciences, telecommunications, insurance, and finance. The investigator develops more accurate and effective methodologies for risk analysis in both univariate and multivariate extreme-value statistics. Progress of the projects in this proposal enhances the collaboration between the investigator and researchers in these fields. The proposed activities also involve teaching graduate students to use extreme-value statistics in their future research. The new methods developed in this proposal are expected to have broader applications as well. For example, they can be used by actuaries to calculate and insure against the probability of rare but financially devastating events, or be employed by statisticians to calculate the required height of sea walls to prevent flooding. They can also be used to tell engineers how strong to build bridges or oil rigs and to model excessively high pollution levels.