Statistica Sinica 27 (2017), 983-996
Abstract: A partial correlation-based variable selection method was proposed for normal linear regression models by Büuhlmann, Kalisch and Maathuis (2010) as an alternative to regularization methods for variable selection. This paper addresses issues related to (a) whether the method is sensitive to the normality assumption, and (b) whether the method is valid when the dimension of predictor increases at an exponential rate in the sample size. To address (a), we study the method for elliptical linear regression models. Our ﬁnding indicates that the original proposal can lead to inferior performance when the marginal kurtosis of predictor is not close to that of normal distribution, and simulation results conﬁrm this. To ensure the superior performance of the partial correlation-based variable selection procedure, we propose a thresholded partial correlation (TPC) approach to select signiﬁcant variables in linear regression models. We establish the selection consistency of the TPC in the presence of ultrahigh dimensional predictors. Since the TPC procedure includes the original proposal as a special case, our results address the issue (b) directly. As a by-product, the sure screening property of the ﬁrst step of TPC is obtained. Numerical examples illustrate that the TPC is comparable to the commonly-used regularization methods for variable selection.
Key words and phrases: Elliptical distribution, model selection consistency, par-tial correlation, partial faithfulness, sure screening property, ultrahigh dimensional linear model, variable selection.