Estimation of prediction error by using <Emphasis Type="Italic">K</Emphasis>-fold cross-validation

Estimation of prediction accuracy is important when our aim is prediction. The training error is an easy estimate of prediction error, but it has a downward bias. On the other hand, K-fold cross-validation has an upward bias. The upward bias may be negligible in leave-one-out cross-validation, but it sometimes cannot be neglected in 5-fold or 10-fold cross-validation, which are favored from a computational standpoint. Since the training error has a downward bias and K-fold cross-validation has an upward bias, there will be an appropriate estimate in a family that connects the two estimates. In this paper, we investigate two families that connect the training error and K-fold cross-validation.