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Statistica Sinica 28 (2018), 2409-2433

EMPIRICAL LIKELIHOOD RATIO TESTS FOR
COEFFICIENTS IN HIGH-DIMENSIONAL
HETEROSCEDASTIC LINEAR MODELS
Honglang Wang 1, Ping-Shou Zhong 2 and Yuehua Cui 2
1 Indiana University-Purdue University Indianapolis
and 2Michigan State University

Abstract: This paper considers hypothesis testing problems for a low-dimensional coefficient vector in a high-dimensional linear model with heteroscedastic variance. Heteroscedasticity is a commonly observed phenomenon in many applications, including finance and genomic studies. Several statistical inference procedures have been proposed for low-dimensional coefficients in a high-dimensional linear model with homoscedastic variance, which are not applicable for models with heteroscedastic variance. The heterscedasticity issue has been rarely investigated and studied. We propose a simple inference procedure based on empirical likelihood to overcome the heteroscedasticity issue. The proposed method is able to make valid inference even when the conditional variance of random error is an unknown function of high-dimensional predictors. We apply our inference procedure to three recently proposed estimating equations and establish the asymptotic distributions of the proposed methods. Simulation studies and real data applications are conducted to demonstrate the proposed methods.

Key words and phrases: Empirical likelihood, heteroscedastic linear models, high-dimensional data, low-dimensional coefficients.

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