Nonparametric estimation of the density of regression errors

Consider the nonparametric regression model Y=m(X) + ε, where the function m is smooth but unknown, and ε is independent of X. An estimator of the density of the error term ε is proposed and its weak consistency is obtained. The strategy used here is based on the kernel estimation of the residuals. Our contribution is twofold. First, we evaluate the impact of the estimation of the regression function m 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 respectively, are proposed. Further, we investigate the asymptotic normality of the error density estimator and its rate-optimality. © 2011 Académie des sciences