Abstract
<jats:p>Motivated by the slow convergence rate of the classical nonparametric kernel density estimator, we study more efficient density and density derivative estimations for the marginal density of nonparametric regression models. In the presence of unknown nonparametric regression function, the proposed density and density derivative estimators can achieve parametric convergence rate, √n, and possess several appealing properties which the classical estimator lacks. In the absence of nonparametric regression function, in the normal case the proposed method performs as well as if we have known the model and estimated the density using maximum likelihood method. Based on the new density estimator, we further propose a more powerful density-based specification test for the nonparametric regression function. Extensive numerical studies show that the proposed density estimator, density derivative estimator, and specification test significantly outperform existing ones.</jats:p>