Empirical likelihood based inference for the derivative of the nonparametric regression function

Abstract

We study statistical inference for the derivative of the nonparametric regression function using local linear model based empirical likelihood. We first derive a normal equation for the derivative through the local linear model and use this equation to construct an empirical likelihood for the derivative. We show that the limiting distribution of the empirical likelihood ratio is a scaled ${\chi }_1^2$ distribution rather than the usual (unscaled) ${\chi }_1^2$ distribution. We use this limiting distribution to construct pointwise confidence intervals for the derivative. Such empirical likelihood ratio confidence intervals are easier to obtain than the normal approximation based confidence intervals. A small simulation study also suggests that they are more accurate.