A note on residual-based empirical likelihood kernel density estimation

Abstract

In general the empirical likelihood method can improve the performance of estimators by including additional information about the underlying data distribution. Application of the method to kernel density estimation based on independent and identically distributed data always improves the estimation in second order. In this paper we consider estimation of the error density in nonparametric regression by residual-based kernel estimation. We investigate whether the estimator is improved when additional information is included by the empirical likelihood method. We show that the convergence rate is not effected, but in comparison to the residual-based kernel estimator we observe a change in the asymptotic bias of the empirical likelihood estimator in first order and in the asymptotic variance in second order. Those changes do not result in a general uniform improvement of the estimation, but in typical examples we demonstrate the good performance of the residual-based empirical likelihood estimator in asymptotic theory as well as in simulations.