Supplemental Materials for : “ The Likelihood Ratio Test in High - Dimensional Logistic Regression Is Asymptotically a Rescaled Chi - Square ”

This document presents the proof of Lemma 6(ii) given in the paper [1]: “The Likelihood Ratio Test in High-Dimensional Logistic Regression Is Asymptotically a Rescaled Chi-Square”. 1 Proof of Lemma 6(ii) We shall prove that V(τ) < τ whenever τ is sufficiently large. Before proceeding, we recall from the main text and [2, Proposition 6.4] that V(τ) := 1 κ E [ Ψ(τZ; b(τ)) ] = 1 κ E [( b(τ)ρ′ ( proxb(τ)ρ (τZ) ))2] , (1) where b(τ) obeys κ = E [Ψ′ (τZ; b(τ))] = 1− E  1 1 + b(τ)ρ′′ ( proxb(τ)ρ (τZ) )  . (2) In what follows, we study the logistic and probit models separately. 1.1 The logistic case Consider the bivariate functions h (b, τ) : = E [ 1 1 + bρ′′ ( proxbρ (τZ) )] , w (b, τ) = E [( ρ′ ( proxbρ (τZ) ))2] , which plays a central role in (1) and (2). In the sequel, we will first analyze these two functions for any b obeying b = c0τ (3) for some constant c0 > 0. The result is this: ∗Department of Statistics, Stanford University, Stanford, CA 94305, U.S.A. †Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, U.S.A. ‡Department of Mathematics, Stanford University, Stanford, CA 94305, U.S.A.