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

许多多变量推理和应用程序都是围绕数据集或其底层分布的中心性和/或极端性发展的。该提案的目标是：（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极端性的影响，以及（ii）证明这些方法在实际应用中的有用性，包括：使用数据深度来构建一个用于研究抑郁和焦虑的自我复杂性的易处理的度量，在同时监测多个风险指标时检测具有极端风险的表现，并对不同的基因组进行分类，以获得更有效的医学治疗。研究结果将推进每个主题的理论基础，并扩大统计对其他领域的适用性。本研究的目的是建立一个全面的多元统计分析框架，有助于发展一种有意义的自我复杂性测量方法，以诊断精神疾病患者，并为此类患者提供更好的医疗服务。这项研究还旨在设计一个有效的阈值系统，用于发出极端风险信号，这对于罕见事件的风险管理应该是有用的，例如航空安全或气候变化引起的灾难性事件。

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.