Sparse linear regression models of high dimensional covariates with non-Gaussian outliers and Berkson error-in-variable under heteroscedasticity

Abstract

The use of penalized least squares based estimations in high dimensional linear regression has been received a great deal of attention during past decades and applied to variable selections under sparsity scenarios. However, it remains unclear that complex data with multiple uncertainties in measurements, such as outliers, error-in-variables and heteroscedasticity, to what extent, can influence these approaches. This paper, here, demonstrates the behaviors of least-squares based approaches in complex data problems, and suggests an alternative approach, Wald-type estimator, relying on theoretical perspectives. Numerical experiments and real data analysis are also presented to illustrate the performance of the different estimation strategies.

Keywords:

• Non-Gaussian uncertainties
• Wald-type estimator
• High dimensional regression
• Sparsity
• Heteroscedasticity

Acknowledgments

The authors wish to express the great gratitude to the editor and the reviewer for their constructive and valuable comments which lead to substantial improvements of the article. We also thank Professor M.J. Hwang for useful suggestions in real data analysis and Dr. C.H. Wu for the assistance in TCGA data retrieving.