From Centrality To Extremity in Multivariate Statistics: Data Depth, Extreme Value Theory and Applications

许多多变量推理和应用都是围绕数据集或其基本分布的中心性和/或终结点而发展的。这项建议的目标是：(I)开发新的非参数统计方法，通过使用数据深度和极值理论来研究数据的中心性和终极性的影响，以及(Ii)展示这些方法在实际应用中的有效性，包括：使用数据深度来构建用于研究抑郁和焦虑的易于处理的自我复杂性度量，在同时监测多个风险度量时检测具有极端风险的表现，以及为更有效的医疗对不同的基因组组进行分类。研究结果将推动每个主题背后的理论，并扩大统计学在其他领域的适用性。研究思路交织在一起，都是为了建立一个全面的多元统计分析方案。这项研究有助于开发一种有意义的自我复杂性测量来诊断精神障碍患者，并为这类患者提供更好的医疗保健。这项研究还旨在设计一种发出极端风险信号的有效门槛系统，该系统应有助于罕见事件的风险管理，如航空安全或气候变化造成的灾难性事件。

Much of multivariate inference and applications evolve around the centrality and/or extremity of the data sets or their underlying distributions. The goals of this proposal are: (i) to develop new nonparametric statistical methodologies for studying the effects of centrality and extremity of data by using data depth and extreme value theory, and (ii) to demonstrate the usefulness of these methodologies in real-life applications, including: using data depth to construct a tractable measure of self-complexity for studying depression and anxiety, detecting performances with extreme risk in the simultaneous monitoring of multiple risk measures, and classifying different genome groups for more effective medical treatments. The research findings would advance the theory underlying each topic and broaden the applicability of statistics to other fields. The lines of investigation are interwoven and are all motivated to build a comprehensive multivariate statistical analysis scheme.This research helps to develop a meaningful self-complexity measure to diagnose patients with psychiatric disorders and provide better health care for such patients. This research also aims to devise an effective threshold system for signaling extreme risks, which should be useful for risk management of rare events, such as in aviation safety or catastrophic events due to climate changes.