An ISIS screening approach involving threshold/partition for variable selection in linear regression

In linear regression, one can select a predictor if the absolute sample correlation between the predictor and the response variable is large. This screening approach is called SIS (sure independence screening) in Fan and Lv (2008). We propose a threshold for SIS and show that using SIS with this threshold can select all important predictors with with probability tending to one as $n \rightarrow \infty$ when $p=O(n^{\delta})$ for some $\delta \in (0, 2)$. For $p$ is of moderate size, we propose an iterative SIS procedure for predictor selection using the proposed threshold. For very large $p$, we propose to include a partitioning step to divide predictors into smaller groups to perform screening. Simulation results show that the first procedure works well for moderate $p$ and the second procedure works well for very large $p$.