Goodness-of-fit test for Gaussian regression with block correlated errors

We propose a procedure to test that the expectation of a Gaussian vector is linear against a nonparametric alternative. We consider the case where the covariance matrix of the observations has a block diagonal structure. This framework encompasses regression models with autocorrelated errors, heteroscedastic regression models, mixed-effects models and growth curves. Our procedure does not depend on any prior information about the alternative. We prove that the test is asymptotically of the nominal level and consistent. We characterize the set of vectors on which the test is powerful and prove the classical √log log (n)/n convergence rate over directional alternatives. We propose a bootstrap version of the test as an alternative to the initial one and provide a simulation study in order to evaluate both procedures for small sample sizes when the purpose is to test goodness of fit in a Gaussian mixed-effects model. Finally, we illustrate the procedures using a real data set.