Incorporating Extra Information in Nonparametric Smoothing

Nonparametric estimation of conditional mean functions has been studied extensively in the literature. This paper addresses the question of how to use extra informations to improve the estimation. Particularly, we consider the situation that the conditional mean functionE(ZX) is of interest and there is an auxiliary variable available which is correlated with bothXandZ. A two-stage kernel smoother is proposed to incorporate the extra information. We prove that the asymptotic optimal mean squared error of the proposed estimator is smaller than that obtained when using the Nadaraya-Watson estimator directly without the auxiliary variable. A simulation study is also carried out to illustrate the procedure.