Finite Sample Performance of Multivariate Location and Scatter Estimators

摘要：本研究为多元位置和散射估计器的性能评估提供了新的方法和理论。本研究旨在基于多元位置和散点估计器的“尾部行为”提出有限样本性能准则，并根据所提出的准则调查和评估现有和新的多元位置和散点估计器的性能，并为应用中的统计实践提供指导。给定一个未知参数的估计量，一个自然的问题是——这个估计量有多好？或者应该如何衡量它的表现？这些问题是统计估计和推断的基础。为了回答这些问题，人们提出并研究了各种绩效标准。在现有的性能准则中，最常用的是Fisher一致性和大样本正态性等渐近方法。这些方法是基于估计量在样本量接近无穷大时的行为。然而，这引起了对实践中渐近结果相关性的一些严重关注，其中样本量总是固定和有限的。因此，研究多元位置和散射估计的有限样本性能不仅具有实际意义，而且在理论上也很有意义。与渐近方法形成鲜明对比的是，本研究研究了固定和有限样本量下多元位置和散射估计的性能。该研究为估计器性能的比较提供了新的方法和有价值的见解，特别是从鲁棒性的角度。我们的有限样本性能测量与鲁棒和非参数统计分析和推理中两个最重要和最有前途的概念的内在联系，有限样本击穿点和数据深度（特别是tukey - donoho半空间深度），将被探索和阐明。结果表明，具有高击穿点或半空间深度的估计器具有显著的有限样本尾性能。这样的研究结果为我们提供了新的见解，加深了我们对击穿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器鲁棒性定量评估的重要作用。该研究对统计学在社会科学、行为科学和生命科学、环境科学和生物科学、工业和经济学等各个学科的应用产生了深远的影响。例如，研究表明，在实践中，对于多变量数据分析中观测值极值的鲁棒性，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位数）应该优于传统的最小二乘估计器（例如，多变量均值）。通过培养研究生和将开发的方法纳入统计课程，这项研究也有利于教育。

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.