Bootstrapping likelihood for model selection with small samples

Akaike’s information criterion (AIC), derived from asymptotics of the maximum likelihood estimator, is widely used in model selection. However, it has a finite-sample bias that produces overfitting in linear regression. To deal with this problem, Ishiguro, Sakamoto, and Kitagawa proposed a bootstrap-based extension to AIC which they called EIC. This article compares model-selection performance of AIC, EIC, a bootstrap-smoothed likelihood cross-validation (BCV) and its modification (632CV) in small-sample linear regression, logistic regression, and Cox regression. Simulation results show that EIC largely overcomes AIC’s overfitting problem and that BCV may be better than EIC. Hence, the three methods based on bootstrapping the likelihood establish themselves as important alternatives to AIC in model selection with small samples.