Adaptive model checking for regressions

Existing model checking methods use the conditional expectation of residual given all predictors as a basis to construct test statistics. However, this strategy involves all predictors even when hypothetical model is of dimension reduction structure. This strategy makes the following disadvantages. First, the sam- pling null distributions of local smoothing-based tests have slow convergence rates to the limiting null distributions and can detect local alternatives distinct from the null at slow rates when the number of predictors is large. Second, the limiting null distributions of empirical process-based tests are intractable. In this project, we propose an adaptive model checking approach for regressions to construct tests that can automatically adapt the dimension reduction struc- tures under the null and general alternative hypothesis accordingly. When the hypothetical model is of a dimension reduction structure such as linear, gener- alized linear or single-index model, adaptive local smoothing tests can detect alternatives distinct from the null at a convergence rate as if there were only one predictor, and the limiting null distributions of adaptive empirical process- based tests are tractable and can still detect local alternatives distinct from the null at parametric rate. Clearly, this research project investigates some important problems in sta- tistical inference when there are many predictors, and expect the new method- ologies and theories developed in the course of the project will have a lasting impact on statistical science.