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

In the common nonparametric regression model with high-dimensional predicto r, several tests for the hypothesis of an additive regression are investiga ted. The corresponding test statistics are based either on the differences between a fit under the assumption of additivity and a fit in the general m odel, or on residuals under the assumption of additivity. For all tests asy mptotic normality is established under the null hypothesis of additivity an d under fixed alternatives with different rates of convergence correspondin g to both cases. These results are used for a comparison of the different m ethods. It is demonstrated that a statistic based on an empirical L-2-dista nce of the Nadaraya-Watson and the marginal integration estimator yields th e (asymptotically) most efficient procedure, if these are compared with res pect to the asymptotic behaviour under fixed and local alternatives. The fi nite-sample properties of the proposed procedures are investigated by means of a simulation study, which qualitatively confirms the asymptotic results .