Asymptotic Optimality for Cₚ, CL, Cross-Validation and Generalized Cross-Validation: Discrete Index Set

Cₚ, CL, cross-validation and generalized cross-validation are useful data-driven techniques for selecting a good estimate from a proposed class of linear estimates. The asymptotic behaviors of these procedures are studied. Some easily interpretable conditions are derived to demonstrate the asymptotic optimality. It is argued that cross-validation and generalized cross-validation can be viewed as some special ways of applying CL. Applications in nearest-neighbor nonparametric regression and in model selection are discussed in detail.