SGER: Statistics of Extremes, with Applications in Financial Time Series

该方案追求尾部独立性的新检验统计量和尾部相关性的新度量的一系列发展，特别是研究了极端相关系数、尾部相关性度量、伽马检验统计量。这些都与多元极值统计的研究有关。在金融应用方面，该提案还将为资产定价和市场数据的极端共同变动指定新的模型，并开发对生态变动更加敏感的新投资组合评估工具。介绍了一种新的极值协移度量方法，包括发展极大稳定过程的统计估计方法。文中还介绍了用马尔可夫过程模型和极大稳定过程模型计算投资组合风险度量的步骤。由于极值相关性的所有定义都依赖于某些极限过程，而且往往涉及到特定参数空间边缘参数的统计检验，因此必须非常小心地获得足够强大的检验。这项提议的目的恰恰是为这个问题找到解决方案。研究人员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。真正的数据将来自保险、金融、电信、气候学、地震学、医学等多个领域。在不同领域的应用中(如上文所述)，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 pursues a series of developments of new test statisticsfor tail independence and of new measures for tail dependence.Particularly, the investigator studies extreme correlationcoefficient, tail dependence measures, gamma test statistics. Theseare related to the study of statistics of multivariate extremes. Infinancial applications, the proposal will also specify new modelsfor asset pricing and extreme co-movements for market data, anddevelop new portfolio evaluation tools more sensitive to theseco-movements. A new measure for extreme co-movement is introduced.The proposal includes the development of statistical estimationmethods for max-stable processes. Procedures of how to calculateportfolio risk measures are also introduced using combined Markovprocess and max-stable process models.As all definitions used for extremal dependence depend on some limitprocedures and often concern statistical testing for parameters atthe edge of a specific parametric space, great care has to be takento obtain tests with sufficient power. The proposal is exactlyaiming at finding a solution for this. The investigator willcarefully compare and contrast new approaches with existing onesusing simulated as well as real data. The real data will come fromareas as diverse as insurance, finance, telecommunications,climatology, seismology, medicine, etc. 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.