Spatial Trimming, with Applications to Robustify Sample Spatial Quantile and Outlyingness Functions, and to Construct a New Robust Scatter Estimator

The spatial multivariate median has a long history as an alternative to the sample mean. Its transformation-retransformation (TR) sample version is affine equivariant, highly robust, and computationally easy. More recently, an entire TR spatial multivariate quantile function has been developed and applied in practice along with related rank functions. However, as quantile levels move farther out, robustness of the TR sample version as measured by breakdown point decreases to zero, a serious limitation in applications such as outlier detection and setting inner 50%, 75%, and 90% quantile regions. Here we introduce a new device, “spatial trimming”, and with it solve two problems of general scope and application: (i) the need for robustification of the TR sample spatial quantile function and its closely related depth, outlyingness, and rank functions, and (ii) the need for a computationally easy, robust, and affine equivariant scatter estimator. Improvements in robustness accomplished by spatial trimming are confirmed by improved breakdown points and illustrated using simulated and actual data. Other applications of spatial trimming are described, and general conclusions and recommendations are provided. AMS 2000 Subject Classification: Primary 62H99. Secondary 62G99