Quotient Correlation, Nonlinear Dependence, and Extreme Dependence Modeling

该提案包括五个重要步骤。首先，研究者开发了新的相关性度量（商相关）并分析了由此产生的检验统计量。重要的是，要尽可能多地基于适当的尾部观测值来开发尾部独立性的检验统计量，而不是基于可能使分析产生偏差的中心观测值。显然，“在哪里合适”的立场的决定必须仔细分析。第二，一个新的措施，极端共同运动的介绍。这种措施允许一个计算概率发生的某些极端事件在未来给定的历史极端运动的基础变量.第三，该提案包括发展的统计估计方法forx-stable过程。这使得人们能够有效地研究clusteredspatial-temporal极端观测。第四，介绍了如何使用马尔可夫过程和最大稳定过程模型计算投资组合风险度量（如无条件或条件风险价值）的过程。第五，极端协同运动的理论分析在实践中具有相当重要的意义。例如，全球金融市场和信用损失数据的溢出概念可以被认为是一个应用领域。该提议的智力价值首先来自于极端相关性的精确测试程序。由于所有使用的定义都依赖于一些极限过程，并且通常涉及特定参数空间边缘参数的统计检验（因此可能是一个非正则估计问题），因此必须非常小心地获得具有一致性的检验。这项建议正是针对这一点寻求解决办法。解决这一问题的几种方法已经公布，然而，到目前为止，似乎还没有明确的获胜方法。研究人员将使用模拟和真实的数据仔细比较和对比新方法与现有方法。真实的数据将来自保险、金融、电信、气候学、地震学、医学等不同领域。除了这些方法的优点和具体应用外，该提案也具有相当广泛的影响。在不同领域的应用中（如上所述），极端风险发挥着重要的科学，社会以及（可能）政治作用。传播新的统计工具，使人们更好地了解联合极端事件的发生，这一点至关重要。这可以很好地实现在新的研究生课程的水平，在期刊上的出版物，旨在更广泛的受众，并在讨论与科学家从其他领域。

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.