Nonparametric kernel estimation of the error density

TL;DR

This paper proposes a nonparametric method to estimate the error density in regression models, analyzing the impact of estimating the regression function and optimizing bandwidth choices for improved accuracy.

Contribution

It introduces an estimator for the error density, evaluates the effect of estimating the regression function, and determines optimal bandwidths for both steps.

Findings

Estimator is weakly consistent

Optimal bandwidths are proposed

Asymptotic normality is established

Abstract

Consider the nonparametric regression model Y=m(X)+E, where the function m is smooth but unknown, and E is independent of X. An estimator of the density of the error term E is proposed and its weak consistency is obtained. The contribution of this paper is twofold. First, we evaluate the impact of the estimation of the regression function on the error density estimator. Secondly, the optimal choices of the first and second step bandwidths used for estimating the regression function and the error density are proposed. Further, we investigate the asymptotic normality of the error density estimator and evaluate its performances in simulated examples.