Multivariate extremes: theory and applications beyond the classical paradigm

多元极端：超越经典范式的理论和应用

• 批准号: 402550-2011
• 负责人: Lysenko, Natalia
• 金额: $ 1.24万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2011
• 资助国家: 加拿大
• 起止时间: 2011-01-01 至 2012-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=488107
• 关键词: Multivariate; extremes; theory; applications; beyond

Multivariate extremes lie at the heart of many fields of application including finance and insurance, civil and reliability engineering, hydrology and environmental sciences. In a risk analysis context, it is not the expected or typical outcomes but the extreme and rare ones that are of greatest concern. The traditional approach to analyzing and modelling of multivariate extremes is based on the asymptotic theory for coordinate-wise maxima, referred to as classical extreme value theory (EVT). Although still relevant for a wide range of problems, this approach suffers from several shortcomings and limitations. This calls for alternative methodological formulations and approaches that go beyond the paradigm of the classical theory. Exploring these alternatives constitutes the main theme of the proposed research program. The first part of the proposal deals with the probabilistic development of multivariate EVT using a geometric approach, initiated in our recent work. In particular, it involves the analysis of multivariate extremes and extreme dependencies in the model and data via the shape of the level sets of the underlying probability density function and the global asymptotic shape of the associated sample clouds. The second part concerns the issues surrounding statistics of multivariate extremes. The main challenge in this context is obvious - it is the very nature of extreme and hence rare events, which leaves either too little or no data to conduct statistical inference. In this situation, asymptotic theory can be used to provide a basis for data extrapolation. The aim is to develop statistical methodology for the analysis of multivariate extremes based on the probabilistic results of the first part of the proposal. The area of multivariate extremes and in particular its statistical estimation side is still in the process of active development, which started just a couple of decades ago. The results and methods to be investigated as part of the proposed research program are seen as a contribution towards maturity of this important and challenging area. The true significance of the proposed work will come in the form of the bridge between mathematical discoveries and their applications to the complex real life problems.

多元极端是许多应用领域的核心，包括金融和保险、土木和可靠性工程、水文和环境科学。在风险分析环境中，最值得关注的不是预期的或典型的结果，而是极端的和罕见的结果。传统的多变量极值分析和建模方法是基于逐坐标极值的渐近理论，即经典极值理论（EVT）。尽管这种方法仍然适用于广泛的问题，但它存在一些缺点和局限性。这需要超越经典理论范式的替代方法论表述和方法。探索这些替代方案构成了拟议研究计划的主题。提案的第一部分涉及使用几何方法的多元EVT的概率发展，这是我们最近的工作中发起的。特别是，它涉及通过潜在概率密度函数的水平集的形状和相关样本云的全局渐近形状来分析模型和数据中的多变量极值和极端依赖性。第二部分是关于多元极端统计的问题。这方面的主要挑战是显而易见的——极端和罕见事件的本质，留给我们的数据要么太少，要么根本没有，无法进行统计推断。在这种情况下，可以使用渐近理论为数据外推提供依据。其目的是根据提案第一部分的概率结果，发展多元极值分析的统计方法。多元极值领域，特别是其统计估计方面，在几十年前才开始活跃发展。作为拟议研究计划的一部分，所调查的结果和方法被视为对这一重要而具有挑战性的领域成熟的贡献。这项工作的真正意义在于，它是数学发现与其应用于复杂现实生活问题之间的桥梁。