Dimension reduction-based significance testing in nonparametric regression

A dimension reduction-based adaptive-to-model test is proposed for significance of a subset of covariates in the context of a nonparametric regression model. Unlike existing locally smoothing significance tests, the new test behaves like a locally smoothing test as if the number of covariates was just that under the null hypothesis and it can detect local alternative hypotheses distinct from the null hypothesis at the rate that is only related to the number of covariates under the null hypothesis. Thus, the curse of dimensionality is largely alleviated when nonparametric estimation is inevitably required. In the cases where there are many insignificant covariates, the improvement of the new test is very significant over existing locally smoothing tests on the significance level maintenance and power enhancement. Simulation studies and a real data analysis are conducted to examine the finite sample performance of the proposed test.