A cautionary note on robust covariance plug-in methods

Many multivariate statistical methods rely heavily on the sample covariance matrix. It is well known though that the sample covariance matrix is highly non-robust. One popular alternative approach for "robustifying" the multivariate method is to simply replace the role of the covariance matrix with some robust scatter matrix. The aim of this paper is to point out that in some situations certain properties of the covariance matrix are needed for the corresponding robust "plug-in" method to be a valid approach, and that not all scatter matrices necessarily possess these important properties. In particular, the following three multivariate methods are discussed in this paper: independent components analysis, observational regression and graphical modeling. For each case, it is shown that using a symmetrized robust scatter matrix in place of the covariance matrix results in a proper robust multivariate method.