Optimal testing for additivity in multiple nonparametric regression

We consider the problem of testing for additivity in the standard multiple nonparametric regression model. We derive optimal (in the minimax sense) nonadaptive and adaptive hypothesis testing procedures for additivity against the composite nonparametric alternative that the response function involves interactions of second or higher orders separated away from zero in L2([0, 1]d)-norm and also possesses some smoothness properties. In order to shed some light on the theoretical results obtained, we carry out a wide simulation study to examine the finite sample performance of the proposed hypothesis testing procedures and compare them with a series of other tests for additivity available in the literature.