Finite Sample Performance of Multivariate Location and Scatter Estimators

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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.