Quotient Correlation, Nonlinear Dependence, and Extreme Dependence Modeling

该建议包括五个重要步骤。首先，研究者开发新的依赖度量（商相关）并分析结果的检验统计量。重要的是要开发尾独立性的检验统计，尽可能多地基于适当的尾部观察，而不是来自中心的观察，这可能会使分析产生偏差。显然，必须仔细分析“哪里合适”的决定。其次，介绍了一种新的极端联合运动测量方法。这种方法允许人们根据潜在变量的极端运动的过去历史来计算未来某些极端事件发生的概率。第三，该建议包括对非稳态过程的统计估计方法的发展。这使得人们能够有效地研究聚类时空极端观测。第四，还介绍了如何使用组合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的过程——例如无条件或条件风险值。第五，极端联合运动的统计分析在实践中具有相当重要的意义。例如，全球金融市场溢出和信贷损失数据的概念可以被认为是一个应用领域。这个建议的智力价值首先源于对极端依赖性的精确测试程序。由于所使用的所有定义都依赖于某些极限过程，并且通常涉及特定参数空间边缘的参数统计检验（因此检验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足够功率的检验。这个建议的目的就是要找到解决这个问题的办法。解决这个问题的几种方法已经发表，然而，到目前为止，似乎还没有明确的制胜方法。研究者将使用模拟和真实数据仔细地将新方法与现有方法进行比较和对比。真实的数据将来自不同的领域，如保险、金融、电信、气候学、地震学、医学等。除了这些方法优点和具体应用之外，该建议还具有相当广泛的影响。在各个领域（如上所述）的应用中，极端风险扮演着重要的科学、社会以及（可能）政治角色。新统计工具的传播有助于更好地理解联合极值的发生，这是非常重要的。这可以在新研究生课程、面向广泛读者的期刊上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.