Consistent variable selection in high dimensional regression via multiple testing

This paper connects consistent variable selection with multiple hypotheses testing procedures in the linear regression model Y=Xβ+ɛ, where the dimension p of the parameter β is allowed to grow with the sample size n. We view the variable selection problem as one of estimating the index set I0⊆{1,…,p} of the non-zero components of β∈Rp. Estimation of I0 can be further reformulated in terms of testing the hypotheses β1=0,…,βp=0. We study here testing via the false discovery rate (FDR) and Bonferroni methods. We show that the set I^⊆{1,…,p} consisting of the indices of rejected hypotheses βi=0 is a consistent estimator of I0, under appropriate conditions on the design matrix X and the control values used in either procedure. This technique can handle situations where p is large at a very low computational cost, as no exhaustive search over the space of the 2p submodels is required.