Nonparametric conditional mean testing via an extreme-type statistic in high dimension

We propose a new test to investigate the conditional mean dependence between a response variable and the corresponding covariates in the high dimensional regimes. The test statistic is an extreme-type one built on the nonparametric method. The limiting null distribution of the proposed extreme type statistic under a mild mixing condition is established. Moreover, to make the test more powerful in general structures we propose a more general test statistic and develop its asymptotic properties. The power analysis of both methods is also considered. In real data analysis, we also propose a new way to conduct the feature screening based on our results. To evaluate the performance of our estimators and other methods, extensive simulations are conducted.