Many statistical concepts are defined in the one-dimensional setting. It is not always clear how to generalize these concepts to higher dimensions. There are many multivariate generalizations of the median which are equivalent to the well-known one-dimensional median in the case of univariate data. Multivariate medians found in the literature lack desirable properties, such as being hard to compute for high dimensional data, or not having rotational invariance. We propose new multivariate estimators, RAD-Median and Zt-Mean, as generalizations of the one-dimensional sample median and trimmed mean, respectively, noting that the median is a special case of the trimmed mean. The parameter t in Zt-Mean controls the trimming proportion. The new estimators are simple and easy to compute and have various desirable properties, such as shift invariance, rotational invariance, and robustness. Simulation results indicate that the RAD-Median has the smallest absolute bias for large sample size and small fraction of outliers, among several other estimators in the literature including L1 median and depth median. Zt-Mean has the smallest mean squared error for some value of t, which depends on the fraction of outliers in the data.
