Multivariate descriptive measures for location and scatter are the foundation of multivariate statistics. Many multivariate statistical methods are based on them. The classical measures of these concepts are moment-based and are estimated by the sample mean vector and the sample covariance matrix. However, these estimators are not robust and are extremely sensitive to outliers. To eliminate or reduce the effects of outliers, we propose general depth-based trimmed means and trimmed covariance matrices. The basic properties of the trimmed means and trimmed covariance matrices are studied. It is shown that these new estimators are not only robust but also efficient, and one can balance robustness and efficiency by trimming percentage in practical applications. Under some regularity conditions, both the trimmed means and trimmed covariance matrices are strongly consistent and asymptotically normal.

Jin Wang is a professor of statistics at Northern Arizona University (NAU), USA. His main research areas include nonparametric multivariate analysis, biostatistics, reliability analysis, probabilistic risk analysis, change-point problems, and asymptotic theory. His contributions to nonparametric multivariate statistics include a nonparametric multivariate kurtosis measure (Wang and Serfling, 2005), a family of kurtosis orderings for multivariate distributions (Wang, 2009), a generalized spread function and a generalized multivariate kurtosis ordering (Wang and Zhou, 2012), a graphical method to compare spread and kurtosis of two multivariate data sets and a new graphical method to assess multivariate normality (Wang, 2019), and so on. Besides theoretical research, Wang is also interested in applications of statistics in various fields. He worked in industry for eight years (1991-1999) and worked on several health-related projects at NAU.