Tests for high-dimensional partially linear regression modelsTests for high-dimensional partially linear regression modelsH. Shi et al.

In this paper, we consider the tests for high-dimensional partially linear regression models. The presence of high-dimensional nuisance covariates and the unknown nuisance function makes the inference problem very challenging. We adopt machine learning methods to estimate the unknown nuisance function and introduce quadratic-form test statistics. Interestingly, though the machine learning methods can be very complex, under suitable conditions, we establish the asymptotic normality of our introduced test statistics under the null hypothesis and local alternative hypotheses. We further propose a power-enhanced procedure to improve the performance of test statistics. Two thresholding determination methods are provided for the proposed power-enhanced procedure. We show that the power enhancement procedure is powerful to detect signals under either sparse or dense alternatives and it can still control the type-I error asymptotically under the null hypothesis. Numerical studies are carried out to illustrate the empirical performance of our introduced procedures.