Inference Problems in Extreme Value Statistics

极值统计中的推理问题

• 批准号: 0604176
• 负责人: Yongcheng Qi
• 金额: --
• 依托单位: University of Minnesota Duluth
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-15 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604176&HistoricalAwards=false
• 关键词: Inference; Problems; Extreme; Value; Statistics

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.

极值理论的研究近年来备受关注。估计罕见事件的概率是人们的主要兴趣之一。这促使许多研究人员在极值统计中开发新的方法。应用极值理论的困难之一是必须仔细选择样本分数，以便估计具有尽可能快的收敛速率，而其偏差可以忽略不计。本提案包括以下五个主题。首先，研究人员提出了一种数据倾斜方法，当底层分布属于其中一个极值分布的吸引域时，构建极端尾部概率的置信区间。所提出的方法有望在覆盖概率方面产生更准确的置信区间，并且对样本分数的选择更加稳健。其次，研究者开发了新的方法来估计二元极值的依赖结构。在二元极值统计中，谱测度等相关结构的估计是一个重要的问题。光谱测量与两个边际极限一起决定了二元极值的极限分布。第三，研究者提出了依赖函数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区间。第四，研究了二元极值统计中谱测度函数和尾相关函数的置信带构造。应用特殊的自举技术来解决这些问题。这允许人们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，提出了具有未知边际分布的高维极值分布的Pickands依赖函数的新估计。极值统计在气象学、水文学、气候学、环境科学、电信、保险、金融等领域都有应用。研究者在单变量和多变量极值统计中开发了更准确和有效的风险分析方法。本提案中项目的进展加强了研究者和研究人员在这些领域的合作。建议的活动还包括教研究生在他们未来的研究中使用极值统计。本提案中开发的新方法也有望有更广泛的应用。例如，它们可以被精算师用来计算罕见但经济破坏性事件的可能性并为其投保，或者被统计学家用来计算防止洪水所需的海堤高度。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。