Abstract: A longstanding problem of existing empirical process-based tests for regressions is that when the number of covariates is greater than one, they either have no tractable limiting null distributions or are not omnibus. To attack this problem, we propose a projection-based adaptive-to-model approach. When the hypothetical model is parametric single-index, the method can fully utilize the dimension reduction model structure under the null hypothesis as if the covariates were one-dimensional such that the martingale transformation-based test can be asymptotically distribution-free. Further, the test can automatically adapt to the underlying model structure such that the test can be omnibus and thus detect alternative models distinct from the hypothetical model at the fastest possible rate in hypothesis testing. The method is examined through simulation studies and is illustrated by a data analysis.

Key words and phrases: Adaptive-to-model test, martingale transformation, model checking, projection pursuit.