Testing for additivity in nonparametric heteroscedastic regression models

This paper introduces a novel hypothesis test for additivity in nonparametric regression models. Inspired by recent advances in the asymptotic theory of analysis of variance when the number of factor levels is large, we develop a test statistic that checks for possible nonlinear relations between the available predictors and the residuals from fitting the additive model. The asymptotic distribution of the test statistic is established under the null and local alternative hypotheses, demonstrating that it can detect alternatives at the rate of $n^{1/4} $n1/4. An advantage over some methods in the literature is that the proposed method maintains its level close to nominal under heteroscedasticity and can be applied to both fixed and random designs. Extensive simulations suggest that the proposed test outperforms competitors for small sample sizes, especially for fixed designs, and performs competitively for larger sample sizes. The proposed method is illustrated with a real dataset.