85ea6fd7a2ca3960d0cf5201933ac998-Supplemental.pdf

Theproof for Lemma A.1 can be found in Theorem 4 in [48]. Similar proof procedure also applies to prove the second inequality in (S.34). Proof by contradiction: suppose that the above second-order condition in (S.42)-(S.43) is false. Thus, in the following, we will ignore the computation process for obtainingξ(θ,γ) and simply view thatξ(θ,γ)is the twice continuously differentiable function of(θ,γ) near (θ,0) and ξθ = ξ(θ=θ,γ=0). Look at (S.62) and note that when = 0andγ = 0, (S.62) is identical to the first two equations in (S.59).