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We analyze the performance of cross-validation (CV) in the density estimationframework with two purposes: (i) risk estimation and (ii) model selection. Themain focus is given to the so-called leave-$p$-out CV procedure (Lpo), where$p$ denotes the cardinality of the test set. Closed-form expressions aresettled for the Lpo estimator of the risk of projection estimators. Theseexpressions provide a great improvement upon $V$-fold cross-validation in termsof variability and computational complexity. From a theoretical point of view,closed-form expressions also enable to study the Lpo performance in terms ofrisk estimation. The optimality of leave-one-out (Loo), that is Lpo with $p=1$,is proved among CV procedures used for risk estimation. Two model selectionframeworks are also considered: estimation, as opposed to identification. Forestimation with finite sample size $n$, optimality is achieved for $p$ largeenough [with $p/n=o(1)$] to balance the overfitting resulting from thestructure of the model collection. For identification, model selectionconsistency is settled for Lpo as long as $p/n$ is conveniently related to therate of convergence of the best estimator in the collection: (i) $p/n\to1$ as$n\to+\infty$ with a parametric rate, and (ii) $p/n=o(1)$ with somenonparametric estimators. These theoretical results are validated by simulationexperiments.

Optimal cross-validation in density estimation with the $L^2$-loss