Test for high dimensional regression coefficients of partially linear models

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.

Keywords:

• High dimensional data
• partially linear models
• generalized U-statistic
• power of test

2000 MSC:

• Primary 62F05
• Secondary 62J05

Acknowledgments

The authors are grateful to the Editor-in-Chief, the Associate Editor and referees for comments, suggestions, and detailed advice that led to the present, vastly improved paper.

Additional information

Funding

The research of Siyang Wang was supported by the Natural Science Foundation of China (No: 11501586). The research of Hengjian Cui was supported by the Natural Science Foundation of China (Nos: 11471223, 11231010, 11071022), BCMIIS, Key project of Beijing Municipal Educational Commission (No. KZ201410028030) and Capacity Building for Sci-Tech Innovation-Fundamental Scientific Research Funds (No. 19530050181).