The existence and asymptotic properties of a backfitting projection algorithm under weak conditions

We derive the asymptotic distribution of a new backfitting procedure for estimating the closest additive approximation to a nonparametric regression function. The procedure employs a recent projection interpretation of popular kernel estimators provided by Mammen, Marron, Turlach and Wand (1997), and the asymptotic theory of our estimators is derived using the theory of additive projections reviewed in Bickel, Klaassen, Ritov, and Wellner (1993). Our procedure achieves the same bias and variance as the oracle estimator based on knowing the other components, and in this sense improves on the method analysed in Opsomer and Ruppert (1997). We provide ‘high level’ conditions independent of the sampling scheme. We then verify that these conditions are satisfied in a regression and a time series autoregression under weak conditions.