On the consistency of cross-validation in kernel nonparametric regression. (English) Zbl 0539.62046

Wong, Wing Hung

Ann. Stat. 11, 1136-1141 (1983).

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.

MSC:

62G05	Nonparametric estimation
62J02	General nonlinear regression
62J05	Linear regression; mixed models

Keywords:

kernel regression estimate; cross-validation; average squared error; Strong consistency; uniform spacing; compact kernel; finite fourth error moment