Repeated weigthed hold-out for variable selection in linear regression

Variable selection criteria in a linear regression model are analyzed, considering consistency and efficiency properties. A Cross-Validation method, which we called Repeated Weigthed Hold-Out, is introduced. This method estimates the Prediction Error, as the well known Repeated Hold-Out, with the difference that, at each iteration, the units extracted in the training-set reduce their probability of a new extraction. This weighting makesthe procedure similar to a repeated K-fold Cross-Validation, in which the units appear with the same frequency in the training sets, obtaining a procedure more flexible than the K-fold CV. It is known that a criterion based exclusively on the estimate of PE is not consistent but efficient, however, using a property of CV, i.e. small training-sets introduce large penalty of the criterion while large training-set introduce small penalty, it is possible to obtain an estimator of the PE inherently penalized. This method dominates several criteria for variable selection, choosing theappropriate training-set size.