Stahel-Donoho estimators with cellwise weights

Abstract

The Stahel–Donoho estimator is defined as a weighted mean and covariance, where each observation receives a weight which depends on a measure of its outlyingness. Therefore, all variables are treated in the same way whether they are responsible for the outlyingness or not. We present an adaptation of the Stahel–Donoho estimator, where we allow separate weights for each variable. By using cellwise weights, we aim to only downweight the contaminated variables such that we avoid losing the information contained in the other variables. The goal is to increase the precision and possibly the robustness, of the estimator. We compare several variants of our proposal and show to what extent they succeed in identifying and downweighting precisely those variables which are contaminated. We further demonstrate that in many situations the mean-squared error of the adapted estimators is lower than that of the original Stahel–Donoho estimator and that this results in better outlier detection capabilities. We also consider some real data examples.