Adaptive-to-Model Hybrid of Tests for Regressions

In model checking for regressions, nonparametric estimation-based tests usually have tractable limiting null distributions and are sensitive to oscillating alternative models, but suffer from the curse of dimensionality. In contrast, empirical process-based tests can, at the fastest possible rate, detect local alternatives distinct from the null model, yet are less sensitive to oscillating alternatives and rely on Monte Carlo approximation for critical value determination, which is costly in computation. We propose an adaptive-to-model hybrid of moment and conditional moment-based tests to fully inherit the merits of these two types of tests and avoid the shortcomings. Further, such a hybrid makes nonparametric estimation-based tests, under the alternatives, also share the merits of existing empirical process-based tests. The methodology can be readily applied to other kinds of data and construction of other hybrids. As a by-product in sufficient dimension reduction field, a study on residual-related central mean subspace and central subspace for model adaptation is devoted to showing when alternative models can be indicated and when cannot. Numerical studies are conducted to verify the powerfulness of the proposed test.