<P> If the model is correctly specified, it can be shown under mild assumptions that the expected value of the MSE for the training set is (n − p − 1) / (n + p + 1) <1 times the expected value of the MSE for the validation set (the expected value is taken over the distribution of training sets). Thus if we fit the model and compute the MSE on the training set, we will get an optimistically biased assessment of how well the model will fit an independent data set . This biased estimate is called the in - sample estimate of the fit, whereas the cross-validation estimate is an out - of - sample estimate . </P> <P> Since in linear regression it is possible to directly compute the factor (n − p − 1) / (n + p + 1) by which the training MSE underestimates the validation MSE, cross-validation is not practically useful in that setting (however, cross-validation remains useful in the context of linear regression in that it can be used to select an optimally regularized cost function). In most other regression procedures (e.g. logistic regression), there is no simple formula to make such an adjustment . Cross-validation is, thus, a generally applicable way to predict the performance of a model on a validation set using computation in place of mathematical analysis . </P> <P> Two types of cross-validation can be distinguished, exhaustive and non-exhaustive cross-validation . </P> <P> Exhaustive cross-validation methods are cross-validation methods which learn and test on all possible ways to divide the original sample into a training and a validation set . </P>

Cross-validation is a robust method of evaluation that can be used for all ie systems