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In this paper we propose a new methodology for testing the parametric formsof the mean and variance functions based on weighted residual empiricalprocesses and their martingale transformations in regression models. Thedimensions of the parameter vectors can be divergent as the sample size goes toinfinity. We then study the convergence of weighted residual empiricalprocesses and their martingale transformation under the null and alternativehypotheses in the diverging dimension setting. The proposed tests based onweighted residual empirical processes can detect local alternatives distinctfrom the null at the fastest possible rate of order $n^{-1/2}$ but are notasymptotically distribution-free. While the tests based on martingaletransformed weighted residual empirical processes can be asymptoticallydistribution-free, yet, unexpectedly, can only detect the local alternativesconverging to the null at a much slower rate of order $n^{-1/4}$, which issomewhat different from existing asymptotically distribution-free tests basedon martingale transformations. As the tests based on the residual empiricalprocess are not distribution-free, we propose a smooth residual bootstrap andverify the validity of its approximation in diverging dimension settings.Simulation studies and a real data example are conducted to illustrate theeffectiveness of our tests.

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