## The Annals of Statistics

### Zonoid trimming for multivariate distributions

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 20 November 2003

https://projecteuclid.org/euclid.aos/1069362382

Digital Object Identifier
doi:10.1214/aos/1069362382

Mathematical Reviews number (MathSciNet)
MR1474078

Zentralblatt MATH identifier
0881.62059

#### Citation

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