High-breakdown estimators of multivariate location and scatter

Abstract

This contribution gives a brief summary of robust estimators of multivariate location and scatter. We assume that the original (uncontaminated) data follow an elliptical distribution with location vector μ and positive definite scatter matrix Σ. Robust methods aim to estimate μ and Σ even though the data has been contaminated by outliers. The well-known multivariate M-estimators can break down when the outlier fraction exceeds 1/(p+1) where p is the number of variables. We describe several robust estimators that can withstand a high fraction (up to 50 %) of outliers, such as the minimum covariance determinant estimator (MCD), the Stahel–Donoho estimator, S-estimators and MM-estimators. We also discuss faster methods that are only approximately equivariant under linear transformations, such as the orthogonalized Gnanadesikan–Kettenring estimator and the deterministic MCD algorithm.