## The Annals of Statistics

### Breakdown Properties of Location Estimates Based on Halfspace Depth and Projected Outlyingness

#### Abstract

We describe multivariate generalizations of the median, trimmed mean and $W$ estimates. The estimates are based on a geometric construction related to "projection pursuit." They are both affine equivariant (coordinate-free) and have high breakdown point. The generalization of the median has a breakdown point of at least $1/(d + 1)$ in dimension $d$ and the breakdown point can be as high as $1/3$ under symmetry. In contrast, various estimators based on rejecting apparent outliers and taking the mean of the remaining observations have breakdown points not larger than $1/(d + 1)$ in dimension $d$.

#### Article information

Source
Ann. Statist. Volume 20, Number 4 (1992), 1803-1827.

Dates
First available in Project Euclid: 12 April 2007

http://projecteuclid.org/euclid.aos/1176348890

JSTOR

Digital Object Identifier
doi:10.1214/aos/1176348890

Mathematical Reviews number (MathSciNet)
MR1193313

Zentralblatt MATH identifier
0776.62031

Subjects
Primary: 62F35: Robustness and adaptive procedures
Secondary: 62G05: Estimation 62H12: Estimation

#### Citation

Donoho, David L.; Gasko, Miriam. Breakdown Properties of Location Estimates Based on Halfspace Depth and Projected Outlyingness. Ann. Statist. 20 (1992), no. 4, 1803--1827. doi:10.1214/aos/1176348890. http://projecteuclid.org/euclid.aos/1176348890.