A robust proposal of estimation for the sufficient dimension reduction problem

In nonparametric regression contexts, when the number of covariables is large, we face the curse of dimensionality. One way to deal with this problem when the sample is not large enough is using a reduced number of linear combinations of the explanatory variables that contain most of the information about the response variable. This leads to the so-called sufficient reduction problem. The purpose of this paper is to obtain robust estimators of a sufficient dimension reduction, that is, estimators which are not very much affected by the presence of a small fraction of outliers in the data. One way to derive a sufficient dimension reduction is by means of the principal fitted components (PFC) model. We obtain robust estimations for the parameters of this model and the corresponding sufficient dimension reduction based on a tau-scale (tau-estimators). Strong consistency of these estimators under weak assumptions of the underlying distribution is proven. The tau-estimators for the PFC model are computed using an iterative algorithm. A Monte Carlo study compares the performance of tau-estimators and maximum likelihood estimators. The results show clear advantages for tau-estimators in the presence of outlier contamination and only small loss of efficiency when outliers are absent. A proposal to select the dimension of the reduction space based on cross-validation is given. These estimators are implemented in R language through functions contained in the package tauPFC. As the PFC model is a special case of multivariate reduced-rank regression, our proposal can be applied directly to this model as well.