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Statistica Sinica 29 (2019), 387-407

REGULARIZATION AFTER RETENTION IN ULTRAHIGH
DIMENSIONAL LINEAR REGRESSION MODELS
Haolei Weng, Yang Feng and Xingye Qiao
Michigan State University, Columbia University and Binghamton University

Abstract: In ultrahigh dimensional setting, independence screening has been both theoretically and empirically proved a useful variable selection framework with low computation cost. In this work, we propose a two-step framework using marginal information in a different fashion than independence screening. In particular, we retain significant variables rather than screening out irrelevant ones. The method is shown to be model selection consistent in the ultrahigh dimensional linear regression model. To improve the finite sample performance, we then introduce a three-step version and characterize its asymptotic behavior. Simulations and data analysis show advantages of our method over independence screening and its iterative variants in certain regimes.

Key words and phrases: Independence screening, Lasso, penalized least square, retention, selection consistency, variable selection.

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