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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 |