## The Annals of Statistics

### Linear hypothesis testing for high dimensional generalized linear models

#### Abstract

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

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

Dates
Revised: July 2018
First available in Project Euclid: 3 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.aos/1564797860

Digital Object Identifier
doi:10.1214/18-AOS1761

Mathematical Reviews number (MathSciNet)
MR3988769

Zentralblatt MATH identifier
07114925

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

#### Citation

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. https://projecteuclid.org/euclid.aos/1564797860

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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.