Linear hypothesis testing for high dimensional generalized linear models

Abstract

This paper is concerned with testing linear hypotheses in highdimensional generalized linear models. To deal with linear hypotheses, we first propose 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 χ 2 distribution with the same degrees of freedom, and under local alternatives, they asymptotically follow non-central χ 2 distributions with the same degrees of freedom and noncentral parameter, provided the number of parameters involved in the test hypothesis grows to ∞ 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.

An Adaptive Test on High-dimensional Parameters in Generalized Linear Models

“…Several new methods ( Ma et al, 2020 ; Shi et al, 2019 ; Sur and Candès, 2019 ; Fei and Li, 2019 ; Zhu et al, 2019 ) have recently been proposed for statistical inference with high-dimensional generalized linear models. However, they mainly focused on related but different questions with different approaches.…”

“… Fei and Li (2019) proposed a multi-sample splitting and averaging method to test a fixed subset of parameters. Shi et al (2019) and Zhu et al (2019) extended the score/Wald/likelihood ratio tests to (non-convex) penalized/constrained regression to test a subset of parameters of size much smaller than the sample size. In principle, due to its data-adaptive feature, aiSPU (with suitable modifications) may be a powerful tool to tackle these related problems, though rigorous investigation is warranted.…”

“…Based on the idea of polyhedral selection, Lee et al (2016) proposed an exact post-selection estimator conditional on the selection event. Meanwhile, many researchers exploit the idea of projection or bias-correction to handle the impact of regularization and high-dimensional nuisance parameters (e.g., Javanmard and Montanari, 2014 ; Van de Geer et al, 2014 ; Zhang and Zhang, 2014 ; Lee et al, 2016 ; Shi et al, 2019 ; Ma et al, 2020 ). In spite of exciting progresses in the last few years, little work has been done to construct more general and powerful tests on high-dimensional regression coefficients in GLMs in the presence of high-dimensional nuisance parameters.…”

Is There a Dose-Response Relationship between Acute Physical Activity and Sleep Length? A Longitudinal Study with Children and Adolescents Living in Sweden

“…Furthermore, the AIC test was also used to test if the variable “time in MVPA” was best defined as continuous or categorical (categories for the quintiles of time in MVPA). The Wald test is a significance test that also can test linear hypotheses [ 23 ]. A Wald test was conducted after the AIC test to test if the relationship between sleep length and time in MVPA departs from linearity.…”

Poisson Regression With Error Corrupted High Dimensional Features

“…is not obvious how to correct this loss function when X i is replaced by W i due to the term e x T i β . (II) As derived in detail in Section 3, the second derivative of the loss function contains random quantities with heavy tailed distributions, therefore the standard Poisson lasso regression, which requires bounded regression function (Shi et al 2019), cannot guarantee to recover the true parameters. In fact, the second derivative turns out to be n´1 ř n i"1 e W T i β´β T varpUqβ rtW i varpUqβu b2´v arpUqu, which has heavy tail because W is not bounded.…”