Nonparametric generalized likelihood ratio test and significant testing for regressions: dimension reduction approaches

Hypothesis testing for regressions is a necessary step in statistical analy- sis. Existing methods have been proved to be powerful when the number of predictors is small. The examples include the nonparametric generalized like- lihood ratio test (NGLRT) for parametric single-index models and the local smoothing-based significance test (LSST) for significant predictors in nonpara- metric models. However, the dimensionality problem causes a big challenge in this research field. Existing model checking methods often use the conditional expectation of residual given all predictors as a basis to construct test statistics. This strategy involves all predictors even when the hypothetical models are of dimension reduction structures and have the following shortcoming. The sam- pling null distributions of local smoothing-based tests have slow convergence rates to the limiting null distributions and thus, the significance level cannot be maintained well when its limiting null distribution is used to determine critical values and resampling approximation is computational intensive. Further, these tests are not sensitive to local alternatives such that they can only detect local alternatives distinct from the null at slow rates when the number of predictors is large. As the examples, both the NLGRT and LSST typically suffer from these problems.

In this project, we propose a bias correction for the NLGRT such that it is asymptotically unbiased. More importantly, an adaptive-to-model construction is applied such that the bias-corrected NLGRT and LSST can fully use the dimension reduction structure under the null hypothesis such that the tests behave like the ones with less number of predictor(s) (it is one for the NLGRT). This can greatly reduce the curse of dimensionality. The method can also be applied to other model checking problems.

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.