Testing additivity by kernel based methods : what is a reasonable test?

In the common nonparametric regression model with high dimensional predictor several tests for the hypothesis of an additive regression are investigated. The corresponding test statistics are either based on the diiferences between a fit under the assumption of additivity and a fit in the general model or based on residuals under the assumption of additivity. For all tests asymptotic normality is established under the null hypothesis of additivity and under fixed alternatives with different rates of convergence corresponding to both cases. These results are used for a comparison of the different methods. It is demonstrated that a statistic based on an empirical L1 - distance of the Nadaraya Watson and the marginal integration estimator yields the asymptotically most efficient procedure if these are compared with respect to the asymptotic behaviour under fixed alternatives.