The Annals of Statistics

Zonoid trimming for multivariate distributions

Gleb Koshevoy and Karl Mosler

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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.

Article information

Ann. Statist., Volume 25, Number 5 (1997), 1998-2017.

First available in Project Euclid: 20 November 2003

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H05: Characterization and structure theory
Secondary: 52A22: Random convex sets and integral geometry [See also 53C65, 60D05] 60F05: Central limit and other weak theorems

Trimmed regions data depth expectile multivariate median quantile


Koshevoy, Gleb; Mosler, Karl. Zonoid trimming for multivariate distributions. Ann. Statist. 25 (1997), no. 5, 1998--2017. doi:10.1214/aos/1069362382.

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