The Annals of Statistics

Linear hypothesis testing for high dimensional generalized linear models

Chengchun Shi, Rui Song, Zhao Chen, and Runze Li

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This paper is concerned with testing linear hypotheses in high dimensional generalized linear models. To deal with linear hypotheses, we first propose the constrained partial regularization method and study its statistical properties. We further introduce an algorithm for solving regularization problems with folded-concave penalty functions and linear constraints. To test linear hypotheses, we propose a partial penalized likelihood ratio test, a partial penalized score test and a partial penalized Wald test. We show that the limiting null distributions of these three test statistics are $\chi^{2}$ distribution with the same degrees of freedom, and under local alternatives, they asymptotically follow noncentral $\chi^{2}$ distributions with the same degrees of freedom and noncentral parameter, provided the number of parameters involved in the test hypothesis grows to $\infty$ at a certain rate. Simulation studies are conducted to examine the finite sample performance of the proposed tests. Empirical analysis of a real data example is used to illustrate the proposed testing procedures.

Article information

Ann. Statist., Volume 47, Number 5 (2019), 2671-2703.

Received: June 2017
Revised: July 2018
First available in Project Euclid: 3 August 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F03: Hypothesis testing
Secondary: 62J12: Generalized linear models

High dimensional testing linear hypothesis likelihood ratio statistics score test Wald test


Shi, Chengchun; Song, Rui; Chen, Zhao; Li, Runze. Linear hypothesis testing for high dimensional generalized linear models. Ann. Statist. 47 (2019), no. 5, 2671--2703. doi:10.1214/18-AOS1761.

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Supplemental materials

  • Supplement to “Linear hypothesis testing for high dimensional generalized linear models”. This supplemental material includes power comparisons with existing test statistics, additional numerical studies on Poisson regression and a real data application, discussions of Conditions (A1)–(A4), some technical lemmas and the proof of Theorem 2.1.