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Testing Regression Coefficients in High-Dimensional and Sparse Settings

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Abstract

In the high-dimensional setting, this article considers a canonical testing problem in multivariate analysis, namely testing coefficients in linear regression models. Several tests for high-dimensional regression coefficients have been proposed in the recent literature. However, these tests are based on the sum of squares type statistics, that perform well under the dense alternatives and suffer from low power under the sparse alternatives. In order to attack this issue, we introduce a new test statistic which is based on the maximum type statistic and magnifies the sparse signals. The limiting null distribution of the test statistic is shown to be the extreme value distribution of type I and the power of the test is analysed. In particular, it is shown theoretically and numerically that the test is powerful against sparse alternatives. Numerical studies are carried out to examine the numerical performance of the test and to compare it with other tests available in the literature.

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Acknowledgements

We appreciate the constructive suggestions from the referees and the editors.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Anhui Normal University, Wuhu, 241002, P. R. China

    Kai Xu & Yan Tian

  2. Center for Quantitative Medicine, Duke-NUS Medical School, National University of Singapore, Singapore, 169856, Singapore

    Qing Cheng

Authors
  1. Kai Xu
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  2. Yan Tian
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  3. Qing Cheng
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Corresponding author

Correspondence to Kai Xu.

Additional information

Supported by the National Natural Science Foundation of China (Grant No. 11901006), the Natural Science Foundation of Anhui Province (Grant No. 1908085QA06) and the Talent Foundation of Anhui Normal University (Grant No. 751811)

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Xu, K., Tian, Y. & Cheng, Q. Testing Regression Coefficients in High-Dimensional and Sparse Settings. Acta. Math. Sin.-English Ser. 37, 1513–1532 (2021). https://doi.org/10.1007/s10114-021-9468-8

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  • DOI: https://doi.org/10.1007/s10114-021-9468-8

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