Direct Estimation of Low-Dimensional Components in Additive Models

Additive regression models have turned out to be a useful statistical tool in analyses of high-dimensional data sets. Recently, an estimator of additive components has been introduced by Linton and Nielsen which is based on marginal integration. The explicit definition of this estimator makes possible a fast computation and allows an asymptotic distribution theory. In this paper an asymptotic treatment of this estimate is offered for several models. A modification of this procedure is introduced. We consider weighted marginal integration for local linear fits and we show that this estimate has the following advantages. (i) With an appropriate choice of the weight function, the additive components can be efficiently estimated: An additive component can be estimated with the same asymptotic bias and variance as if the other components were known. (ii) Application of local linear fits reduces the design related bias.