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的中心性和/或极值或其底层分布而发展的。本提案的目标是：(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值性的影响，以及（ii）证明这些方法在实际应用中的有用性，包括：利用数据深度构建可处理的自复杂性度量，用于研究抑郁和焦虑，在多个风险度量同时监测中检测极端风险表现，以及对不同基因组组进行分类，以便更有效地进行医学治疗。研究结果将推动每个主题的理论基础，并将统计学的适用性扩大到其他领域。调查线是相互交织的，都是为了建立一个全面的多元统计分析方案。本研究有助于开发一种有意义的自我复杂性测量方法来诊断精神疾病患者，并为这些患者提供更好的医疗保健。本研究还旨在设计一个有效的极端风险信号阈值系统，用于罕见事件的风险管理，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.