Abstract Partially linear models attract much attention to investigate the association between predictors and the response variable when the dependency on some predictors may be nonlinear. However, the hypothesis test for significance of predictors is still challenging, especially when the number of predictors is larger than sample size. In this paper, we reconsider the test procedure of Zhong and Chen (2011) when regression models have nonlinear components, and propose a generalized U-statistic for testing the linear components of the high dimensional partially linear models. The asymptotic properties of test statistic are obtained under null and alternative hypotheses, where the effect of nonlinear components should be considered and thus is different from that in linear models. Through simulation studies, we demonstrate good finite-sample performance of the proposed test in comparison with the existing methods. The practical utility of our proposed method is illustrated by a real data example.

中文翻译：

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检验部分线性模型的高维回归系数

摘要 当对某些预测变量的依赖性可能是非线性时，偏线性模型在研究预测变量与响应变量之间的关联方面备受关注。然而，预测变量显着性的假设检验仍然具有挑战性，尤其是当预测变量的数量大于样本量时。在本文中，当回归模型具有非线性分量时，我们重新考虑了 Zhong and Chen (2011) 的测试程序，并提出了一个广义 U 统计量来测试高维部分线性模型的线性分量。检验统计量的渐近性质是在零假设和替代假设下获得的，其中应考虑非线性分量的影响，因此与线性模型中的不同。通过模拟研究，与现有方法相比，我们证明了所提出测试的良好有限样本性能。我们提出的方法的实际效用通过一个真实的数据例子来说明。