Likelihood-based cross-validation score for selecting the smoothing parameter in maximum penalized likelihood estimation

Maximum penalized likelihood estimation is applied in non(semi)-parametric regression problems, and enables us exploratory identification and diagnostics of nonlinear regression relationships. The smoothing parameter λ controls trade-off between the smoothness and the goodness-of-fit of a function. The method of cross-validation is used for selecting λ, but the generalized cross-validation, which is based on the squared error criterion, shows bad behavior in non-normal distribution and can not often select reasonable λ. The purpose of this study is to propose a method which gives more suitable λ and to evaluate the performance of it. A method of simple calculation for the delete-one estimates in the likelihood-based cross-validation (LCV) score is described. A score of similar form to the Akaike information criterion (AIC) is also derived. The proposed scores are compared with the ones of standard procedures by using data sets in literatures. Simulations are performed to compare the patterns of selecting λ and overall goodness-of-fit and to evaluate the effects of some factors. The LCV scores by the simple calculation provide good approximations to the exact one if λ is not extremely small. Furthermore the LCV scores by the simple calculation have little risk of choosing extremely small λ and make it possible to select λ adaptively. They have the effect of reducing the bias of estimates and provide better performance in the sense of overall goodness-of-fit. These scores are useful especially in the case of small sample size and in the case of binary logistic regression.