Statistica Sinica 14(2004), 1199-1208

PROJECTION-BASED AFFINE EQUIVARIANT

MULTIVARIATE LOCATION ESTIMATORS WITH THE

BEST POSSIBLE FINITE SAMPLE BREAKDOWN POINT

Yijun Zuo

Michigan State University

Abstract: The sample mean has long been used as an estimator of a location parameter in statistical data analysis and inference. Though attractive from many viewpoints, it suffers from an extreme sensitivity to outliers. The median has been adopted as a more robust location estimator in one dimension, it will not break down even if up to half of the data points are ``bad''. Another desirable property of the median is that it does not depend on the underlying measurement scale and coordinate system. Clearly, multivariate analogues of the univariate median are practically desirable and theoretically interesting. Among proposed analogues, only the spatial median (the L$_1$-median) and the coordinate-wise median have a breakdown point as high as that of the univariate median. These estimates, however, lack the affine equivariance property. Affine equivariant analogues exist, but their breakdown points decrease as dimension increases. We propose a class of projection-based affine equivariant multivariate location estimators. There are estimators in this class that possess a breakdown point (with respective to a definition slightly weaker than the usual one) as high as that of the univariate median, free of dimension. Compared with the existing best breakdown point affine equivariant location estimators, these estimators can, in some cases, resist up to $10\%$ more contamination in a data set without break down. Computing issues are briefly addressed.

Key words and phrases: Affine equivariance, breakdown point, location estimator, multivariate median, projection pursuit methodology, robustness.