2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/210765We 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 (1993). The result is complementary to that of Yang (2005) and Yang (2006).Statistics Theorytext