Likelihood-based cross-validation is a statistical tool for selecting a density estimate based on n i.i.d. observations from the true density among a collection of candidate density estimators. General examples are the selection of a model indexing a maximum likelihood estimator, and the selection of a bandwidth indexing a nonparametric (e.g. kernel) density estimator. In this article, we establish asymptotic optimality of a general class of likelihood based cross-validation procedures (as indexed by the type of sample splitting used, e.g. V-fold cross-validation), in the sense that the cross-validation selector performs asymptotically as well (w.r.t. to the Kullback-Leibler distance to the true density) as an optimal benchmark model selector which depends on the true density. Crucial conditions of our theorem are that the size of the validation sample converges to infinity, which excludes leave-one-out cross-validation, and that the candidate density estimates are bounded away from zero and infinity. We illustrate these asymptotic results and the practical performance of likelihood based cross-validation for the purpose of bandwidth selection with a simulation study.
Statistical Methodology | Statistical Theory
van der Laan, Mark J.; Dudoit, Sandrine; and Keles, Sunduz, "Asymptotic Optimality of Likelihood Based Cross-Validation" (February 2003). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 125.