4011

BY LEO BREIMAN

The Annals of Statistics

1996

24

6

2350]2383

In model selection, usually a ‘‘best’’ predictor is chosen from a collection mˆ?, s.4 of predictors where mˆ?, s. is the minimum least-squares predictor in a collection Us of predictors. Here s is a complexity parameter; that is, the smaller s, the lower dimensionalrsmoother the models in Us. If L is the data used to derive the sequence mˆ?, s.4, the procedure is called unstable if a small change in L can cause large changes in mˆ?, s.4. With a crystal ball, one could pick the predictor in mˆ?, s.4 having minimum prediction error. Without prescience, one uses test sets, crossvalidation and so forth. The difference in prediction error between the crystal ball selection and the statistician’s choice we call predictive loss. For an unstable procedure the predictive loss is large. This is shown by some analytics in a simple case and by simulation results in a more complex comparison of four different linear regression methods. Unstable procedures can be stabilized by perturbing the data, getting a new predictor sequence mˆX ?, s.4 and then averaging over many such predictor sequences.