CAREER: Statistical Depth Functions and their Applications

统计深度函数作为一种有前途的工具在鲁棒和非参数多元数据分析和推理中越来越受到追捧。本课题将对这些功能及其应用进行系统深入的研究。该项目的目标包括：1)为从业者选择深度函数及其诱导估计量建立基础并提供具体指导；2)引入深度相关的实际推理程序；3)深化和扩展现有深度应用，同时将深度方法推向新的领域；4)开发快速准确的深度函数和相关程序的实际计算算法和工具包。5)将研究活动纳入教育过程，并让代表性不足的少数民族参与拟议的活动。基于排序的简单一维统计在一维数据分析中发挥了如此重要的作用，多年来人们一直在寻找它们的多维类似物，但没有得到完全令人满意的结果。将这些一维统计数据（如中位数）扩展到更高的维度是困难的，因为没有一种自然且明确的方法来对多维观测值进行完全排序或排序。统计深度函数被证明是一种非常有前途的多维观测排序工具。深度函数的主要思想是从“最深”点提供多维观测的“中心向外”排序。多维数据排序并不是深度函数的唯一应用。深度函数为我们带来了多维探索性数据分析和推理的新视角，并已被证明在从工业工程到生物医学科学等学科中具有重要的应用。然而，对理论和方法的深入研究仍处于初级阶段，一些根本性问题有待解决，包括：1)深度函数在许多领域以各种各样的形式被特别地引入，而不考虑它们是否符合任何特定的标准集，也不考虑构造它们的一般机制；2)存在大量的深度诱导估计器，在它们之间的选择几乎没有指导，这对潜在用户和设计软件包的人来说都是一个障碍；3)计算深度函数和相关程序是具有挑战性的——如果没有快速和准确的算法，深度方法的进一步发展可能会受到阻碍；4)深度相关的推理程序还没有得到普遍的发展，深度函数的应用还没有进一步深化、扩展和推向新的领域；最后，5)深度方法还没有整合到教育过程中。该提案将解决所有这些问题。该项目将1)将基于排序的一维统计程序和方法的应用领域和方法论优势扩展到更高维度的背景下，2)促进深度理论和方法论领域的发现和理解，3)推进基于深度方法论的非参数和稳健的多元探索性数据分析。4)促进启发性教学和热情学习，同时扩大代表性不足群体的参与；5)建立一个强大的统计研究和教育计划，涉及深度社区的研究人员，培养来自不同学科的本科生和研究生；6)为PI建立一个坚实的基础，为研究和教育做出综合贡献。

Statistical depth functions have become increasingly pursued as a promising tool in robust and nonparametric multivariate data analysis and inference. This project is to conduct a systematic and thorough study of these functions and their applications. The objectives of this project include: 1)to establish bases and provide specific guidance for selection of depth functions and their induced estimators for practitioners, 2) to introduce depth associated practical inference procedures, 3) to deepen and extend existing depth applications while pushing depth methodology to new frontiers, 4) to develop fast and accurate algorithms and toolkits for the practical computing of depth functions and associated procedures, and 5) to incorporate research activities into the educational process and to engage the participation of underrepresented minorities in the proposed activities. Simple one-dimensional statistics based on ordering have played such an important role in one-dimensional data analysis and their multi-dimensional analogues have been sought for years, without completely satisfactory results. The extension to higher dimensions of these one-dimensional statistics, such as the median, is difficult because there is no natural and unambiguous method of fully ordering or ranking multi-dimensional observations. Statistical depth functions are proving to be a very promising tool for ordering multi-dimensional observations. The main idea of depth functions is to provide from the "deepest" point a "center-outward" ordering of multi-dimensional observations. Multi-dimensional data ordering is not the only application of depth functions though. Depth functions have brought us new perspectives towards multidimensional exploratory data analysis and inference, and have been proven to have significant applications in disciplines ranging from industrial engineering to biomedical sciences. Research in depth theory and methodology, however, is still in its preliminary stage and a number of fundamental issues are yet to be addressed including: 1) depth functions have been introduced ad hoc in many areas, and in great variety, without regards as to whether they meet any particular set of criteria and without regard to a general mechanism to construct them, 2) a large number of depth induced estimators exist with little guidance for choosing among them, an obstacle both, to potential users and to those designing software packages, 3) computing depth functions and associated procedures is challenging-- without fast and accurate algorithms further developments of depth methodology can be hampered, 4) depth-associated inference procedures have not been developed in general, and applications of depth functions are yet to be further deepened, broadened, and pushed to new frontiers: and finally, 5) depth methodology has not been integrated into the educational process. The proposal will address all these issues. The project will 1) extend the areas of application and methodological advantages of one-dimensional statistical procedures and methods based on ordering to the higher-dimensional context, 2) stimulate discovery and understanding within the field of depth theory and methodology, 3) advance nonparametric and robust multivariate exploratory data analysis built on the depth methodology, 4) promote inspired teaching and enthusiastic learning while broadening the participation of underrepresented groups, 5) establish a strong research and education program in statistics involving researchers in the depth community and train undergraduate and graduate students from various disciplines, and 6) build for the PI a firm foundation for a lifetime of integrated contributions to research and education.