Discussion of \Sure Independence Screening for Ultra-High Dimensional Feature Space"
Runze Li · 2008
Screening. Consider a regression model, E(y|x) = η(x β), and assume that x has its characteristic function E(exp(itx) = exp(it μx)φ(t T Σxt). That is, x follows an elliptical distribution with mean μx, covariance matrix Σx and characteristic generator φ(·) (Fang, Kotz and Ng, 1990). Let z = Σ−1/2 x (x− μx). Then E(x− μx)y = E(x− μx)η(x β) = E{Σ1/2 x zη(β Σ x z + β Σ x μx)}. (1) Take Γ to be an orthogonal matrix whose first row is β Σ 1/2 x / √ β Σxβ, and let v = (v1, · · · , vp) = Γz. Then (1) becomes