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 dierences between a t under the assumption of additivity and a t 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 xed alternatives with dierent rates of convergence corresponding to both cases. These results are used for a comparison of the dierent methods. It is demonstrated that a statistic based on an empirical L 2 distance of the Nadaraya Watson and the marginal integration estimator yields the (asymptotically) most ecient procedure, if these are compared with respect to the asymptotic behaviour under xed alternatives.

In this article we highlight the main differences of available methods for the analysis of regression functions that are probably additive separable. We first discuss definition and interpretation of the most common estimators in practice. This is done by explaining the different ideas of modeling behind each estimator as well as what the procedures are doing to the data. Computational aspects are mentioned explicitly. The illustrated discussion concludes with a simulation study on the mean squared error for different marginal integration approaches. Next, various test statistics for checking additive separability are introduced and accomplished with asymptotic theory. Based on the asymptotic results under hypothesis as well as under the alternative of non additivity we compare the tests in a brief discussion. For the various statistics, different smoothing and bootstrap methods we perform a detailed simulation study. A main focus in the reported results is directed on the (non-) reliability of the methods when the covariates are strongly correlated among themselves. Again, a further point are the computational aspects. We found that the most striking differences lie in the different pre-smoothers that are used, but less in the different constructions of test statistics. Moreover, although some of the observed differences are strong, they surprisingly can not be revealed by asymptotic theory. 1

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Economic capital, Regulatory capital, Nonperforming loan, Recovery, Loss given default, C51, G11, G18, G33,

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