In this paper, we consider tests for ultrahigh-dimensional partially linear regression models. The presence of ultrahigh-dimensional nuisance covariates and 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 test statistics' performance. Two thresholding determination methods are provided for the power-enhanced procedure. We show that the power-enhanced 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.

翻译：本文考虑超高维部分线性回归模型的检验问题。超高维干扰协变量和未知干扰函数的存在使得推断问题极具挑战性。我们采用机器学习方法估计未知干扰函数，并引入二次型检验统计量。有趣的是，尽管机器学习方法可能非常复杂，但在适当条件下，我们建立了所提出的检验统计量在原假设和局部备择假设下的渐近正态性。我们进一步提出了一种增强功效的程序以改进检验统计量的性能，并为此增强程序提供了两种阈值确定方法。研究表明，增强功效程序能够有效检测稀疏或稠密备择假设下的信号，同时能在原假设下渐进控制第一类错误。数值实验验证了所提出方法的实证效果。