Zonoid trimming for multivariate distributions
The Annals of Statistics · 1997 · Volume 25 · Issue 5
Abstract
A family of trimmed regions is introduced for a probability distribution in Euclidean d-space. The regions decrease with their parameter $\alpha$, from the closed convex hull of support (at $\alpha = 0$) to the expectation vector (at $\alpha = 1$). The family determines the underlying distribution uniquely. For every $\alpha$ 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.
Keywords
52A22, 60F05, 62H05, data depth, expectile, multivariate median, quantile, Trimmed regions
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