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An important question in constructing Cross Validation (CV) estimators of thegeneralization error is whether rules can be established that allow "optimal"selection of the size of the training set, for fixed sample size $n$. We definethe {\it resampling effectiveness} of random CV estimators of thegeneralization error as the ratio of the limiting value of the variance of theCV estimator over the estimated from the data variance. The variance and thecovariance of different average test set errors are independent of theirindices, thus, the resampling effectiveness depends on the correlation and thenumber of repetitions used in the random CV estimator. We discuss statisticalrules to define optimality and obtain the "optimal" training sample size as thesolution of an appropriately formulated optimization problem. We show that in abroad class of loss functions the optimal training size equals half of thetotal sample size, independently of the data distribution. We optimally selectthe number of folds in $k$-fold cross validation and offer a computationalprocedure for obtaining the optimal splitting in the case of classification(via logistic regression). We substantiate our claims both, theoretically andempirically.

Optimality of Training/Test Size and Resampling Effectiveness of Cross-Validation Estimators of the Generalization Error