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Stepwise Searching for Feature Variables in High-Dimensional Linear Regression

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We investigate the classical stepwise forward and backward search methods for selecting sparse models in the context of linear regression with the number of candidate variables p greater than the number of observations n. Two types of new information criteria BICP and BICC are proposed to serve as the stopping rules in the stepwise searches, since the traditional information criteria such as BIC and AIC are designed for the cases with p LASSO selector. The consistency of the stepwise search is investigated when both $n$ and p tend to infinity. We show that a stepwise forward addition followed by a stepwise backward deletion, both controlled by a version of BICP , leads to a consistent estimated model under the sparse Riesz condition.

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