A robust proposal of estimation for the sufficient dimension reduction problem TEST

YOHAI; VICTOR J.; MARÍA EUGENIA SZRETTER NOSTE; BERGESIO. A

TEST

SPRINGER

Lugar: Berlin; Año: 2021 vol. 30 p. 758 - 773

1133-0686

In non-parametric regression contexts, when the number of covariables is large, weface the curse of dimensionality. One way to deal with this problem when the sample is not largeenough, is using a reduced number of linear combinations of the explanatory variables that containmost of the information about the response variable. This leads to the so called sucient reductionproblem. The purpose of this paper is to obtain robust estimators of a sucient dimension reduction,that is, estimators which are not very much aected by the presence of a small fraction ofoutliers in the data. One way to derive a sucient dimension reduction is by means of the PrincipalFitted Components (PFC) model. We obtain robust estimations for the parameters of this modeland the corresponding sucient dimension reduction based on a 􀀀scale (􀀀estimators). Strongconsistency of these estimators under weak assumptions of the underlying distribution is proven.The 􀀀estimators for the PFC model are computed using an iterative algorithm. A Monte Carlostudy compares the performance of 􀀀estimators and maximum likelihood estimators. The resultsshow clear advantages for 􀀀estimators in the presence of outlier contamination and only smallloss of eciency when outliers are absent. A proposal to select the dimension of the reductionspace based on cross{validation is given. These estimators are implemented in R language throughfunctions contained in the package tauPFC. As the PFC model is a special case of multivariatereduced-rank regression, our proposal can be applied directly to this model as well.