6 Sure Independence Screening for Ultra - High Dimensional Feature Space ∗

High dimensionality is a growing feature in many areas of contemporary statistics. Variable selection is fundamental to high-dimensional statistical modeling. For problems of large or huge scale pn, computational cost and estimation accuracy are always two top concerns. In a seminal paper, Candes and Tao (2007) propose a minimum l1 estimator, the Dantzig selector, and show that it mimics the ideal risk within a logarithmic factor log pn. Their innovative procedure and remarkable result are challenged when the dimensionality is ultra high: the factor log pn can be large and their uniform uncertainty condition can fail. Motivated by these concerns, in this paper we introduce the concept of sure screening and propose a fast and straightforward method via iteratively thresholded ridge regression, called Sure Independence Screening (SIS), to reduce high dimensionality to a relatively large scale dn, say below sample size. An appealing special case of SIS is the componentwise regression. In a fairly general asymptotic framework, SIS is shown to possess the sure screening property for even exponentially growing dimensionality. With ultra-high dimensionality reduced accurately to below sample size, variable selection becomes much easier and can be accomplished by some refined lower-dimensional methods that have oracle properties. Depending on the scale of dn, one can use, for example, the Dantzig selector or Lasso, the fine method of SCAD-penalized least squares in Fan and Li (2001), or the adaptive Lasso in Zou (2006). Short title: Sure Independence Screening AMS 2000 subject classifications: Primary 62J99; secondary 62F12