Assessing prediction error at interpolation and extrapolation points

Common model selection criteria, such as AIC and its variants, are based on in-sample prediction error estimators. However, in many applications involving predicting at interpolation and extrapolation points, in-sample error does not represent the relevant prediction error. In this paper new prediction error estimators, tAI and Loss(w_t) are introduced. These estimators generalize previous error estimators, however are also applicable for assessing prediction error in cases involving interpolation and extrapolation. Based on these prediction error estimators, two model selection criteria with the same spirit as AIC and Mallow’s C_p are suggested. The advantages of our suggested methods are demonstrated in a simulation and a real data analysis of studies involving interpolation and extrapolation in linear mixed model and Gaussian process regression.