Linear hypothesis testing for high dimensional generalized linear models
Chengchun Shi, Rui Song, Zhao Chen, Runze Li
The Annals of Statistics · 2019 · Volume 47 · Issue 5
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.
Keywords
62F03, 62J12, High dimensional testing, likelihood ratio statistics, linear hypothesis, score test, Wald test
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