Regression, Prediction and Shrinkage

Summary The fit of a regression predictor to new data is nearly always worse than its fit to the original data. Anticipating this shrinkage leads to Stein-type predictors which, under certain assumptions, give a uniformly lower prediction mean squared error than least squares. Shrinkage can be particularly marked when stepwise fitting is used: the shrinkage is then closer to that expected of the full regression rather than of the subset regression actually fitted. Preshrunk predictors for selected subsets are proposed and tested on a number of practical examples. Both multiple and binary (logistic) regression models are considered.