Quotient Correlation, Nonlinear Dependence, and Extreme Dependence Modeling

该提案包括五个重要步骤。首先，研究人员开发了新的相关性度量(商相关)，并分析了由此产生的测试统计。重要的是，尽可能多地基于适当的尾部观测来开发尾部独立性的测试统计，而不是基于可能会对分析产生偏差的中心观测。显然，必须仔细分析“在哪里做合适的”的决定。第二，引入了一项极端联合运动的新措施。这一衡量标准允许人们在给定潜在变量极端运动的过去历史的情况下，计算未来发生某些极端事件的概率。第三，该建议包括发展最大稳定过程的统计估计方法。这使得人们能够有效地研究集群的时空极端观测。第四，介绍了组合马尔可夫过程和极大稳定过程模型计算投资组合风险度量--如无条件风险价值和条件风险价值的步骤。第五，极端共同运动的统计分析在实践中具有相当重要的意义。例如，全球金融市场溢出效应和信用损失数据的概念可以被视为一个应用领域。该提议的理论价值首先来自于对极端依赖的精确测试程序。由于所使用的所有定义都依赖于一些极限过程，并且通常涉及对特定参数空间边缘的参数的统计检验(因此，检验可能是一个非正规估计问题)，因此必须非常小心地获得具有足够能力的检验。这项提议正是为了找到解决这一问题的办法。解决这一问题的几个程序已经公布，然而，到目前为止，似乎还没有明确的取胜方法。研究人员将利用模拟和真实数据仔细地将新方法与现有方法进行比较。真实数据将来自保险、金融、电信、气候学、地震、医学等多个领域。除了这些方法的优点和具体应用之外，该提议也具有相当广泛的影响。在不同领域的应用中(如上所述)，极端风险在科学、社会以及(可能)政治方面发挥着重要作用。传播新的统计工具有助于更好地理解联合极端的发生，这是非常重要的。这可以在新的研究生课程、针对更广泛受众的期刊上发表的出版物以及与其他领域的科学家的讨论中很好地实现。

The proposal consists of five important steps. First, theinvestigator develops new dependence measures (quotient correlation)and analyze the resulting test statistics. It is important todevelop test statistics for tail independence based as much aspossible on appropriate tail observations and less on observationsfrom the center which may bias the analysis. Clearly, the decisionon "where does appropriate" stands has to be carefully analyzed.Second, a new measure for extreme co-movement is introduced. Thismeasure allows one to calculate probabilities of occurrences ofcertain extreme events in the future given the past history of theextreme movement of underlying variables. Third, the proposalincludes the development of statistical estimation methods formax-stable processes. This allows one to efficiently study clusteredspatial-temporal extreme observations. Fourth, procedures of how tocalculate portfolio risk measures -- such as unconditional orconditional Value at Risk -- are also introduced using combinedMarkov process and max-stable process models. Fifth, the statisticalanalysis of extreme co-movements is of considerable importance inpractice. For example, the notion of spillover in global financialmarkets and credit loss data can be thought of as one area ofapplication.The intellectual merit of the proposal in a first instance stemsfrom a precise testing procedure for extremal dependence. As alldefinitions used depend on some limit procedures and often concernstatistical testing for parameters at the edge of a specificparametric space (hence possibly testing is a non-regular estimationproblem), great care has to be taken to obtain tests with sufficientpower. The proposal is exactly aiming at finding a solution forthis. Several procedures for tackling this problem have beenpublished, however, up to now, no clear winning approach seems toexist. The investigator will carefully compare and contrast newapproaches with existing ones using simulated as well as real data.The real data will come from areas as diverse as insurance, finance,telecommunications, climatology, seismology, medicine, etc. Beyondthese methodological merits and specific applications, the proposalalso has a considerable broad impact. Throughout applications indiverse fields (like above), extreme risks play an importantscientific, societal as well as (possibly) political role. Thedissemination of new statistical tools leading to a betterunderstanding of the occurrence of joint extremes is of greatimportance. This can be very well achieved at the level of newgraduate courses, publications in journals aimed at a broadenaudience and in discussion with scientists from other fields.