Multidimensional Depth Functions, Multidimensional Generalized L-Statistics, and Related Procedures
多维深度函数、多维广义 L 统计量及相关过程
基本信息
- 批准号:9705209
- 负责人:
- 金额:$ 9.58万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:1997
- 资助国家:美国
- 起止时间:1997-07-15 至 2001-06-30
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
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.
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。建立了统一并扩展了目前文献中关于深度函数的几个例子的一般理论。在多维数据点按“深度”从中心向外排序的概念基础上,提出并研究了多维位置、分布、分位数、秩等传统一维样本统计的相应概念。在这个框架中,研究了诸如多维l统计、秩统计和这些统计的广义形式等统计数据(扩展了研究者在一维情况下的先前工作)。进一步研究了“等高线”的相关概念。统计数据在鲁棒性标准(例如,分解点)、等方差(与几何结构相关)、计算易用性、概念一致性(与相关的人口概念)和理论可追溯性方面表现良好。本研究使用和进一步开发的工具包括功能分析和u统计方法。特别关注的应用环境包括稳健和非参数回归以及方差分析。本研究开发了改进的多维数据分析方法。例如,对于数据点的“云”,人们希望知道“中心”位于何处。可以取这些点的平均值,也可以设法确定一个受数据云极值影响较小的“中点”。同样,数据云的其他代表性特性需要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。本研究对这些问题进行了系统的探讨,并提出了新的研究方法。这种汇总统计数据可以通过几个易于解释的数字来传达数据云的主要特征,从而使人们能够在传统统计报告的范围内充分描述数据。而视觉方法失去其有效性的维度大于三个,总结方法开发在本研究同样适用于任何数量的维度。在现代社会和国家战略关注的各个领域进行的非常复杂的数据收集活动中,越来越多地出现多维数据集。这项研究导致了简化和减少这种复杂性的工具。此外,本研究还开发了一种方法,将数据解释为来自目标人群的样本,并试图对其进行统计推断。例如,解决了这样的数据究竟是用来估计什么的问题。发展这些新的统计方法所需的基本数学进展也作为本研究的一部分完成。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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Robert Serfling其他文献
On Liu’s simplicial depth and Randles’ interdirections
- DOI:
10.1016/j.csda.2016.02.002 - 发表时间:
2016-07-01 - 期刊:
- 影响因子:
- 作者:
Robert Serfling;Yunfei Wang - 通讯作者:
Yunfei Wang
Depth functions in nonparametric multivariate inference
- DOI:
10.1090/dimacs/072/01 - 发表时间:
2003 - 期刊:
- 影响因子:0
- 作者:
Robert Serfling - 通讯作者:
Robert Serfling
On masking and swamping robustness of leading nonparametric outlier identifiers for univariate data
- DOI:
10.1016/j.jspi.2015.02.002 - 发表时间:
2015-07-01 - 期刊:
- 影响因子:
- 作者:
Shanshan Wang;Robert Serfling - 通讯作者:
Robert Serfling
Depth-based nonparametric description of functional data, with emphasis on use of spatial depth
- DOI:
10.1016/j.csda.2016.07.007 - 发表时间:
2017-01-01 - 期刊:
- 影响因子:
- 作者:
Robert Serfling;Uditha Wijesuriya - 通讯作者:
Uditha Wijesuriya
Robert Serfling的其他文献
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{{ truncateString('Robert Serfling', 18)}}的其他基金
Multivariate Depth and Quantile Functions: Foundations and Applications
多元深度和分位数函数:基础和应用
- 批准号:
1106691 - 财政年份:2011
- 资助金额:
$ 9.58万 - 项目类别:
Continuing Grant
Nonparametric Outlyingness and Descriptive Measures in Multivariate and General Data Settings
多元和一般数据设置中的非参数异常性和描述性测量
- 批准号:
0805786 - 财政年份:2008
- 资助金额:
$ 9.58万 - 项目类别:
Standard Grant
Nonparametric and Robust Multivariate Analysis via Quantile Functions
通过分位数函数进行非参数和稳健的多元分析
- 批准号:
0103698 - 财政年份:2001
- 资助金额:
$ 9.58万 - 项目类别:
Continuing Grant
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