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.
Highlights
Let ε1, . . . , εn denote a sample of independent random variables with cumulative distribution function F and density f
In general the empirical likelihood method can improve the performance of estimators by including additional information about the underlying data distribution
We investigate whether the estimator is improved when additional information is included by the empirical likelihood method
Summary
Let ε1, . . . , εn denote a sample of independent random variables with cumulative distribution function F and density f. For this estimator Kiwitt et al [11] showed that both asymptotic bias and variance of F n are different in first order in comparison to the residual-based empirical distribution Fn. The incorporation of the additional information can lead to an improved estimator; in contrast to the i.i.d.-case considered by Qin and Lawless [16], it does not in all cases. In the paper at hand we consider the empirical likelihood density estimator f n based on the residuals and investigate whether in comparison to the residualbased kernel estimator fn the asymptotic bias and variance change and whether an improvement of the estimation can be achieved.
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