Empirical likelihood ratio tests for non-nested model selection based on predictive losses

We propose an empirical likelihood ratio (ELR) test for comparing any two supervised learning models, which may be nested, non-nested, overlapping, misspecified, or correctly specified. The test compares the prediction losses of models based on the cross-validation. We determine the asymptotic null and alternative distributions of the ELR test for comparing two nonparametric learning models under a general framework of convex loss functions. However, the prediction losses from the cross-validation involve repeatedly fitting the models with one observation left out, which leads to a heavy computational burden. We introduce an easy-to-implement ELR test which requires fitting the models only once and shares the same asymptotics as the original one. The proposed tests are applied to compare additive models with varying-coefficient models. Furthermore, a scalable distributed ELR test is proposed for testing the importance of a group of variables in possibly misspecified additive models with massive data. Simulations show that the proposed tests work well and have favorable finite-sample performance compared to some existing approaches. The methodology is validated on an empirical application.