Estimating the Error Rate of a Prediction Rule: Improvement on Cross-Validation

We construct a prediction rule on the basis of some data, and then wish to estimate the error rate of this rule in classifying future observations. Cross-validation provides a nearly unbiased estimate, using only the original data. Cross-validation turns out to be related closely to the bootstrap estimate of the error rate. This article has two purposes: to understand better the theoretical basis of the prediction problem, and to investigate some related estimators, which seem to offer considerably improved estimation in small samples.