Web - based Supplementary Materials for “ Score test variable screening ” by Sihai Dave Zhao and Yi Li Web Appendix A : Theoretical properties of score test screening

Theoretical justifications can be easier to derive for score test-based screening compared to Wald test-based screening. The main task is to find a finite-sample bound for P{|U j (0)| ≥ γn}, which can often be done by applying Bernstein-type inequalities. In contrast, Wald testbased screening requires deriving non-asymptotic tail bounds for the marginal estimators, which can be considerably more involved. We will give a sufficient condition that, under certain assumptions, will guarantee sure screening and false positive control. We assume throughout that the covariates have mean 0 and variance 1. Let uj (βj) be the limiting marginal estimating equation, such that U j (βj) → u M j (βj). Assumption 1 There exists some constant c1 > 0 such that minj∈M |u M j (0)| ≥ c1n −κ with 0 < κ < 1/2.