Normal approximation rate of the kernel smoothing estimator in a partial linear model

By establishing the asymptotic normality for the kernel smoothing estimator <(beta)over cap>(n) of the parametric components beta in the partial linear model Y = X'beta + g(T) + epsilon, P. Speckman (1988, J. Roy. Statist. Sec. Ser. B 50, 413-456) proved that the usual parametric rate n(-1/2) is attainable under the usual "optimal" bandwidth choice which permits the achievement of the optimal nonparametric rate for the estimation of the nonparametric component g. In this paper we investigate the accuracy of the normal approximation for <(beta)over cap>(n) and find that, contrary to what we might expect, the optimal Berry-Esseen rate n(-1/2) is not attainable unless g is undersmoothed, that is, the bandwidth is chosen with faster rate of tending to zero than the "optimal" bandwidth choice. (C) 1999 Academic Press.