The influence function of Stahel-Donoho type methods for robust covariance estimation and

Consider a p-dimensional sample X = (x1,..., xn) of size n. In this paper we will concentrate on Stahel-Donoho type estimators of covariance. By this, we mean estimators based on the Stahel-Donoho outlyingness r(xi,X), defined as follows (Stahel, 1981; Donoho, 1982): r(xi,X) = sup a∈Rp ∣∣∣∣atxi −m(atX)s(atx) where m(.) and s(.) are univariate robust estimators of location and scale. In order to obtain robust estimates of the covariance matrix, we want to concentrate on those data points with small outlyingness. We consider two options. A first approach consists of downweighting all observations according to their outlyingness. We will call this estimator weighted Stahel-Donoho (SDw) from now on. Several choices for the weighting function have been proposed, see Maronna and Yohai (1995), Zuo et al. (2004) and Gervini (2002). A second approach was proposed in Hubert et al. (2005). A proportion 0 < α < 1/2 is chosen. Only the (1 − α)n observations with smallest outlyingness are used in the estimation. We will cal