Nonparametric kernel estimation of the error density

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.