  In variable selection, most existing screening methods focus on marginal
effects and ignore dependence between covariates. To improve the performance of
selection, we incorporate pairwise effects in covariates for screening and
penalization. We achieve this by studying the asymptotic distribution of the
maximal absolute pairwise sample correlation among independent covariates. The
novelty of the theory is in that the convergence is with respect to the
dimensionality $p$, and is uniform with respect to the sample size $n$.
Moreover, we obtain an upper bound for the maximal pairwise R squared when
regressing the response onto two different covariates. Based on these extreme
value results, we propose a screening procedure to detect covariates pairs that
are potentially correlated and associated with the response. We further combine
the pairwise screening with Sure Independence Screening and develop a new
regularized variable selection procedure. Numerical studies show that our
method is very competitive in terms of both prediction accuracy and variable
selection accuracy.
