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Zonoid trimming for multivariate distributions. (English) Zbl 0881.62059
Summary: 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.

##### MSC:
 62H05 Characterization and structure theory for multivariate probability distributions; copulas 52A22 Random convex sets and integral geometry (aspects of convex geometry) 60F05 Central limit and other weak theorems
##### Keywords:
data depth; expectile; multivariate median; quantile; trimmed regions
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