zbMATH — the first resource for mathematics

On the consistency of cross-validation in kernel nonparametric regression. (English) Zbl 0539.62046
Summary: For the nonparametric regression model \(Y(t_ i)=\theta(t_ i)+\epsilon(t_ i)\) where \(\theta\) is a smooth function to be estimated, \(t_ i's\) are nonrandom, \(\epsilon(t_ i)'s\) are i.i.d. errors, this paper studies the behavior of the kernel regression estimate \[ {\hat \theta}(t)=[\sum^{n}_{j=1}K(\lambda^{-1}(t_ j-t))Y(t_ j)]/[\sum^{n}_{j=1}K(\lambda^{-1}(t_ j-t))] \] when \(\lambda\) 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.

62G05 Nonparametric estimation
62J02 General nonlinear regression
62J05 Linear regression; mixed models
Full Text: DOI