Normal Approximation Rate of the Kernel Smoothing Estimator in a Partial Linear Model

Abstract

By establishing the asymptotic normality for the kernel smoothing estimatorβnof the parametric componentsβin the partial linear modelY=X′β+g(T)+ε, P. Speckman (1988,J. Roy. Statist. Soc. Ser. B50, 413–456) proved that the usual parametric raten−1/2is attainable under the usual “optimal” bandwidth choice which permits the achievement of the optimal nonparametric rate for the estimation of the nonparametric componentg. In this paper we investigate the accuracy of the normal approximation forβnand find that, contrary to what we might expect, the optimal Berry–Esseen raten−1/2is not attainable unlessgis undersmoothed, that is, the bandwidth is chosen with faster rate of tending to zero than the “optimal” bandwidth choice.