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

DMS 9705209 Serfling 为了给多维数据的统计分析提供更坚实的基础，本研究发展了统计深度函数的一般理论。 一般理论的发展，统一和扩展目前在文献中的深度功能的几个例子。 基于多维数据点按“深度”的中心向外排序的概念，本研究制定并研究了多维位置、散布、分位数、秩等传统一维样本统计量的相应概念。 在这个框架中，统计，如多维L-统计，秩统计，和这些统计的广义形式，进行了调查（扩展以前的工作的调查人员为一维的情况下）。 此外，相应的概念的“轮廓”进行了研究。 开发了关于鲁棒性标准（例如，分解点），等方差（相关的几何结构），计算方便，概念的一致性（与相关的人口概念），和理论tractability.Tools使用，并进一步发展，为这项研究包括功能分析和U-统计方法。 受到特别关注的应用环境包括稳健的非参数回归和方差分析。 本研究开发了用于分析多维数据的改进方法。 例如，对于数据点的“云”，人们希望了解“中心”的位置。 人们可以取这些点的平均值，或者可以试图定义一个受数据云末端影响较小的“中间点”。 类似地，数据云的其他代表性特征需要被定义为已经用于分析简单一维数据的概念的类似物或扩展。 本研究系统地处理这些问题，并开发新的方法付诸实践。 这种汇总统计数据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，从而使人们能够在传统统计报告的范围内充分描述数据。 虽然视觉方法失去了他们的有效性，超过三个维度，总结方法在这项研究中开发的适用于任何数量的维度。 在现代社会和国家战略问题的各个领域目前进行的非常复杂的数据收集活动中，日益出现多层面数据集。 这项研究导致简化和减少这种复杂性的工具。 此外，这项研究开发的方法，解释的数据，但一个目标人口的样本，其中一个试图作出统计推断。 例如，这类数据究竟在估算什么，这一问题得到了解决。 作为本研究的一部分，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.