| As a measure of center of a multivariate distribution, the spatial multivariate median has a long history as an alternative to the sample mean. Its transformation-retransformation (TR) sample version is computationally easy, affine equivariant, and highly robust. However, as the quantile level is chosen farther out from the center, robustness of the TR sample version as measured by breakdown point decreases to zero, a serious limitation in applications such as outlier detection, for example. In this talk our main focus is to robustify the sample spatial outlyingness functions through a new device "Spatial Trimming". The improvements in robustness of the TR spatial outlyingness accomplished by spatial trimming are confirmed by an improved masking breakdown point and illustrated using simulated and actual data. Several other applications of spatial trimming, including a new robust Mahalanobis distance, have been noted.
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