On the Consistency of Cross-Validation in Kernel Nonparametric Regression

Abstract

For the nonparametric regression model Y(ti) = θ(ti) + (ti) where θ is a smooth function to be estimated, ti's are nonrandom, (ti)'s are i.i.d. errors, this paper studies the behavior of the kernel regression estimate θ̂(t) = [ n j=1K (tj - t/λ) Y(tj) ] / [ sumn j=1 K (tj - t/λ) ] when λ is chosen by cross-validation on the average squared error. Strong consistency in terms of the average squared error is established for uniform spacing, compact kernel and finite fourth error moment.