Some hypothesis testing problems with high-dimensional data

In this project, we investigate three testing problems for regression models when the dimensionality of the covariate vector is large, even larger than the sample size. The problems are: the two-sample Behrens-Fisher problem; model checking for parametric regression models; and testing for the existence of random effects in an ANOVA mixed model. Unlike existing tests in the literature, the new test for the two-sample Behrens-Fisher problem is scale-invariant. For the models with a dimension reduction structure, we suggest a dimension reduction test for the parametric regression models, and a difference-based and a trace-based test for the existence of random effects. There appears to be no previous research discussed in the literature for the last two problems with large dimensionality.

Clearly, this research project investigates some important problems in statistical inference when there are many predictors, and the new methodologies and theories developed in the course of the project will have a lasting impact on statistical science.