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Fan and Lv (2008) proposed the path-breaking theory of sure independencescreening (SIS) and an iterative algorithm (ISIS) to effectively reduce thepredictor dimension for further variable selection approaches. Fan et al.(2009) extended ISIS to generalized linear models and introduced the VanillaISIS (Van-ISIS) algorithm, allowing selected predictors to be screened out inupcoming iterations. The success of SIS depends on its sure screening property,which was obtained by Fan and Lv (2008) under the marginal correlationassumption. However, despite wide applications of ISIS and Van-ISIS in variousscientific fields, their sure screening properties have not been proved duringthe past decade. To fill this gap, we prove the sure screening properties ofthree different types of iterative algorithms for linear models without relyingon the marginal correlation assumption, where ISIS and Van-ISIS can be regardedas two special cases of them.

On the sure screening properties of iteratively sure independence screening algorithms