Model checking for high-dimensional parametric regressions: the conditionally studentized test

This paper studies model checking for general parametric regression models with no dimension reduction structures on the high-dimensional vector of predictors. Using existing test as an initial test, this paper combines the sample-splitting technique and conditional studentization approach to construct a COnditionally Studentized Test(COST). Unlike existing tests, whether the initial test is global or local smoothing-based, and whether the dimension of the predictor vector and the number of parameters are fixed, or diverge at a certain rate as the sample size goes to infinity, the proposed test always has a normal weak limit under the null hypothesis. Further, the test can detect the local alternatives distinct from the null hypothesis at the fastest possible rate of convergence in hypothesis testing. We also discuss the optimal sample splitting in power performance. The numerical studies offer information on its merits and limitations in finite sample cases. As a generic methodology, it could be applied to other testing problems.