Multidimensional Depth Functions, Multidimensional Generalized L-Statistics, and Related Procedures

DMS 9705209 Serfling为了加强多维数据统计分析的基础，本研究发展了统计深度函数的一般理论。发展了一般理论，统一和扩展了目前文献中关于深度函数的几个例子。基于多维数据点按深度向中心向外排序的概念，本研究提出并研究了多维位置、分布、分位数、等级等传统一维样本统计量的相应概念。在这个框架中，研究了多维L统计量、等级统计量以及这些统计量的推广形式(扩展了调查者先前对一维案件的工作)。此外，还研究了相应的“等值线”概念。统计学在稳健性标准(例如，崩溃点)、等变性(与几何结构相关)、计算简易性、概念一致性(与相关总体概念)和理论可操作性方面表现良好。本研究使用并进一步发展的工具包括函数分析方法和U-统计方法。受到特别关注的应用背景包括稳健和非参数回归和方差分析。这项研究开发了分析多维数据的改进方法。例如，对于数据点的“云”，人们希望对“中心”的位置有一种感觉。人们可以取这些点的平均值，或者可以寻求定义一个受数据云末端影响较小的“中间点”。同样，数据云的其他代表性功能需要定义为简单一维数据分析中已使用的概念的类比或扩展。本研究对这些问题进行了系统的研究，并提出了新的实践方法。这种汇总统计使数据云的主要特征能够通过几个易于解释的数字来传达，从而使人能够在传统统计报告的范围内充分描述数据。虽然视觉方法在维度大于3的情况下会失效，但这项研究中开发的总结方法同样适用于任何数量的维度。在现代社会和战略性国家关切的各个领域开展的非常复杂的数据收集活动中，多维数据集日益增多。这项研究导致了简化和降低这种复杂性的工具。此外，这项研究开发了一些方法，将数据解释为目标人群的样本，并试图对其进行统计推断。例如，解决了这样的数据究竟估计了什么的问题。作为这项研究的一部分，还完成了开发这些新统计方法所需的基本数学进展。

DMS 9705209 Serfling In order to provide strengthened foundations for statistical analysis of multidimensional data, this research develops a general theory of statistical depth functions. General theory is developed which unifies and extends the few examples of depth functions presently in the literature. Based on the notion of center-outward ordering of multidimensional data points by "depth," corresponding notions of multidimensional location, spread, quantiles, ranks, and other traditional one-dimensional sample statistics are formulated and studied in this research. In this framework, statistics such as multidimensional L-statistics, rank statistics, and generalized forms of these statistics, are investigated (extending previous work of the investigator for the one-dimensional case). Further, corresponding notions of "contours" are investigated. Statistics are developed which perform well overall with respect to robustness criteria (e.g., breakdown points), equivariance (relevant to the geometric structure), computational ease, conceptual consistency (with associated population notions), and theoretical tractability.Tools used, and further developed, for this research include functional analytic and U-statistic methods. Application contexts receiving special attention include robust and nonparametric regression and analysis of variance. This research develops improved methods for analyzing multidimensional data. For a "cloud" of data points, one wishes to have a sense of where the "center" is located, for example. One can take the average of the points, or one can seek to define a "middle point" that is less influenced by the extremities of the data cloud. Similarly, other representative features of the data cloud need to be defined as analogues or extensions of concepts already in use for analysis of simple one-dimensional data. This research systematically treats such issues and develops new methods to be put into practice. Such summary statistics enable the main featur es of a data cloud to be conveyed by means of a few easily interpretable numbers, thus enabling one to describe the data adequately within the confines of a conventional statistical report. Whereas visual methods lose their effectiveness for dimensions greater than three, the summarizing methods developed in this research apply equally well for any number of dimensions. Multidimensional data sets are arising increasingly in the very complex data-gathering activities now pursued in the various arenas of modern society and strategic national concern. This research leads to tools for simplification and reduction of this complexity. Further, this study develops methods for interpreting the data as but a sample from a target population about which one seeks to make statistical inferences. For example, the question of what exactly is being estimated by such data is addressed. Basic mathematical advances needed for development of these new statistical methods are also accomplished as part of this research.