On the Consistency of Cross-Validation in Kernel Nonparametric Regression

The Annals of Statistics · 1983 · Volume 11 · Issue 4

Abstract

For the nonparametric regression model $Y(t_i) = \theta(t_i) + \varepsilon(t_i)$ where $\theta$ is a smooth function to be estimated, $t_i

#39;s are nonrandom, $\varepsilon(t_i)

#39;s are i.i.d. errors, this paper studies the behavior of the kernel regression estimate $\hat{\theta}(t) = \big\lbrack \sum^n_{j=1}K \big(\frac{t_j - t}{\lambda}\big) Y(t_j) \big\rbrack / \big\lbrack\ sum^n_{j=1} K \big(\frac{t_j - t}{\lambda}\big) \big\rbrack$ 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.

Keywords

62G05, consistency, cross-validation, kernel estimate, Nonparametric regression

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