Cross-validation for comparing multiple density estimation procedures

We demonstrate the consistency of cross-validation for comparing multiple density estimators using simple inequalities on the likelihood ratio. In nonparametric problems, the splitting of data does not require the domination of test data over the training/estimation data, contrary to Shao [Shao, J., 1993. Linear model selection by cross-validation. J. Amer. Statist. Assoc. 88, 486-494]. The result is complementary to that of Yang [Yang, Y., 2007. Consistency of cross-validation for comparing regression procedures, Ann. Statist. 35, 2450-2473; Yang, Y., 2006. Comparing learning methods for classification. Statist. Sinica 16, 635-657].