Finite Sample Performance of Multivariate Location and Scatter Estimators

标题：多元位置和散布估计量的有限样本性能摘要：这项研究在多变量位置和散布估计的性能评估中发展了新的方法和理论。本研究旨在提出基于其“尾部行为”的多变量定位和散布估计量的有限样本性能准则，调查和评估现有和新的多变量定位和散布估计量相对于建议准则的性能，并为应用中的统计实践提供指导。给出一个未知参数的估计器，一个自然的问题--估计器有多好？或者，人们应该如何衡量它的表现？这些问题是统计估计和推断中的基本问题。为了回答这些问题，人们提出并研究了各种绩效标准。在现有的性能指标中，包括Fisher相合性和大样本正态分布在内的渐近方法是最常用的。这些方法都是基于样本大小接近无穷大时估计者的行为。然而，这引起了人们对渐近结果在实践中的相关性的一些严重关注，在实践中，样本大小总是固定和有限的。因此，研究多元位置和散布估计的有限样本性能不仅具有重要的现实意义，而且具有重要的理论意义。与渐近方法形成鲜明对比的是，本文研究了固定样本量和有限样本量下的多变量定位和散布估计量的性能。这项研究发展了新的方法和对估计器性能比较的有价值的见解，特别是从稳健性的角度。我们的有限样本性能度量与稳健和非参数统计分析和推断中最关键和最有前途的两个概念--有限样本故障点和数据深度(特别是Tukey-Donoho半空间深度)--之间的内在联系将被探索和说明。结果表明，具有较高故障点或半空间深度的估计具有显著的有限样本尾部性能。研究中的这类发现为我们提供了新的见解，加深了我们对故障点和半空间深度概念的理解，并确立了有限样本尾部行为作为估计量稳健性的定量评估的重要作用。这项研究对社会科学、行为学和生命科学、环境和生物科学、工业和经济学等不同学科的统计应用产生了深远的影响。例如，研究表明，在多变量数据分析中，相对于传统的最小二乘估计(例如，多变量均值)，在多变量数据分析中，具有吸引人的尾部性能的位置估计器(例如，基于深度的多变量中值)应该更好地具有稳健性。这项研究还通过培训研究生和将已开发的方法纳入统计课程而使教育受益。

TITLE:Finite Sample Performance of Multivariate Location and Scatter EstimatorsABSTRACT:This research develops new methodology and theory in the performance evaluation of multivariate location and scatter estimators. The research is to propose finite sample performancecriteria of multivariate location and scatter estimators based on their ``tail behavior'', to investigate and assess the performance of existing and new multivariate location and scatter estimators with respect to the proposed criteria, and to provide guides to statistical practices in applications. Given an estimator of some unknown parameter, a natural questionis -- how good is the estimator? or how should one measure its performance? These questions are fundamental in statistical estimation and inference. To answer these questions, various performance criteria have been proposed and studied. Among the existing performance criteria, asymptotic approaches including Fisher consistency and large sample normality are the most prevalent ones. These approaches have been based on the behavior of the estimator as the sample size approaches infinity. This, however, raises some serious concern about the relevancy of the asymptotic results in practice, where the sample size is always fixed and finite. The study of the finite sample performance of multivariate location and scatter estimators thus is not only practically significant but also theoretically interesting. In sharp contrast with the asymptotic approaches, in this research the performance of multivariate location and scatter estimators is studied for fixed and finite sample size. The research develops new methodology and valuable insights into the comparison of estimator performance particularly from robustness standpoint. Inherent connections of our finite sample performance measures with two most crucial and promising notions in the robust and nonparametric statistical analysis and inference, the finite sample breakdown point and the data depth (especiallyTukey-Donoho halfspace depth), are to be explored and illuminated. It is to be shown that the estimators with high breakdown point or halfspace depth possess remarkable finite sample tail performance. Findings like this in the research offer new insights into and deepen our understanding of the notions of breakdown point and halfspace depth, and establish the important role of the finite sample tail behavior as a quantitative assessment of robustness of estimators. The research has profound impact on statistical applications in various disciplines of sciences such as social, behavior and life sciences, environmental and biology sciences, industry, and economics. The research suggests, for example, that location estimators with appealing tail performance (e.g. depth-based multivariate medians) should be preferred in practiceto traditional least-squares estimators (e.g. the multivariate mean) for robustness against the influence of extremities of observations in multivariate data analysis. The research also benefits education through the training of graduate students and the incorporation of the developed methodology in statistics courses.