Weighted residual empirical processes, martingale transformations and model checking for regressions

In this paper we propose a new methodology for testing the parametric forms of the mean and variance functions based on weighted residual empirical processes and their martingale transformations in regression models. The dimensions of the parameter vectors can be divergent as the sample size goes to infinity. We then study the convergence of weighted residual empirical processes and their martingale transformation under the null and alternative hypotheses in the diverging dimension setting. The proposed tests based on weighted residual empirical processes can detect local alternatives distinct from the null at the fastest possible rate of order $n^{-1/2}$ but are not asymptotically distribution-free. While the tests based on martingale transformed weighted residual empirical processes can be asymptotically distribution-free, yet, unexpectedly, can only detect the local alternatives converging to the null at a much slower rate of order $n^{-1/4}$, which is somewhat different from existing asymptotically distribution-free tests based on martingale transformations. As the tests based on the residual empirical process are not distribution-free, we propose a smooth residual bootstrap and verify the validity of its approximation in diverging dimension settings. Simulation studies and a real data example are conducted to illustrate the effectiveness of our tests.