  This text is a survey on cross-validation. We define all classical
cross-validation procedures, and we study their properties for two different
goals: estimating the risk of a given estimator, and selecting the best
estimator among a given family. For the risk estimation problem, we compute the
bias (which can also be corrected) and the variance of cross-validation
methods. For estimator selection, we first provide a first-order analysis
(based on expectations). Then, we explain how to take into account second-order
terms (from variance computations, and by taking into account the usefulness of
overpenalization). This allows, in the end, to provide some guidelines for
choosing the best cross-validation method for a given learning problem.
