Zonoid trimming for multivariate distributions

A family of trimmed regions is introduced for a probability distribution in Euclidean d-space. The regions decrease with their parameter α, from the closed convex hull of support (at α = 0) to the expectation vector (at α = 1). The family determines the underlying distribution uniquely. For every α the region is affine equivariant and continuous with respect to weak convergence of distributions. The behavior under mixture and dilation is studied. A new concept of data depth is introduced and investigated. Finally, a trimming transform is constructed that injectively maps a given distribution to a distribution having a unique median.