A cautionary note on robust covariance plug-in methods

The sample covariance matrix, which is well known to be highly nonrobust, plays a central role in many classical multivariate statistical methods. A popular way of making such multivariate methods more robust is to replace the sample covariance matrix with some robust scatter matrix. The aim of this paper is to point out that multivariate methods often require that certain properties of the covariance matrix hold also for the robust scatter matrix in order for the corresponding robust plug-in method to be a valid approach, but that not all scatter matrices possess the desired properties. Plug-in methods for independent components analysis, observational regression and graphical modelling are considered in more detail. For each case, it is shown that replacing the sample covariance matrix with a symmetrized robust scatter matrix yields a valid robust multivariate procedure.