following similar arguments in proving the negligibility of C$_{2}$$_{n}$ . Hence C$_{3}$$_{n}$ = O$_{p}$ ( n − 1 / $^{2}$) = o$_{p}$ (1). This ends the proof of Lemma A.5. □ The next two lemmas establish the (uniform) convergence of G$_{n}$ ( u, ˆ θ$_{n}$ ) and Δ − 1 n ( ˆ θ$_{n}$ ) to G ( u, θ$_{0}$ ) and Δ − $^{1}$( θ$_{0}$ ), respectively. Lemma A.6 Under Assumptions 3.1-3.3, we have Proof of Lemma A.6: The proof follows directly from the ULLN of Newey and McFadden (1994). □ Lemma A.7 Under Assumptions 3.1-3.2, we have Proof of Lemma A.7: The proof follows from the ULLN of Newey and McFadden (1994) and the continuous mapping theorem. □ Now, we are ready to proceed with the proofs of our main theorems. Proof of Theorem 1: By a straightforward decomposition, we have By Lemmas A.4-A.7, we have that 38 = − √$_{n}$$_{(}$ ˆ θ$_{n}$ − θ$_{0}$ ) $^{′}$E [ g ( X, θ$_{0}$ ) ∂g ( X, θ$_{0}$ ) ∂θ ′ ] √$_{n}$$_{(}$ ˆ θ$_{n}$ − θ$_{0}$ ) + o$_{p}$ (1) = O$_{p}$ (1) , sup u ∈ Π ∣ G$_{n}$ ( u, ˆ θ$_{n}$ ) − G ( u, θ$_{0}$ ) ∣ = o$_{p}$ (1) . Δ − 1 n ( ˆ θ$_{n}$ ) = Δ − $^{1}$( θ$_{0}$ ) + o$_{p}$ (1) . ˆ R p $_{n}$( u ) = 1 √$_{n}$ n ∑ i =1 ε$_{i}$ ( ˆ θ$_{n}$ ) ( 1 { q ( X$_{i}$, ˆ θ$_{n}$ ) ≤ u } − g $^{′}$( X$_{i}$, ˆ θ$_{n}$ )Δ − 1 n ( ˆ θ$_{n}$ ) G$_{n}$ ( u, ˆ θ$_{n}$ ) = ˆ R$_{n}$ ( u ) − G ′ $_{n}$( u, ˆ θ$_{n}$ )Δ − 1 n ( ˆ θ$_{n}$ ) 1 √$_{n}$ n ∑ i =1 ε$_{i}$ ( ˆ θ$_{n}$ ) g ( X$_{i}$, ˆ θ$_{n}$ ) := ˆ R$_{n}$ ( u ) − G ′ $_{n}$( u, ˆ θ$_{n}$ )Δ − 1 n ( ˆ θ$_{n}$ ) ˆ S$_{n}$. ˆ R p $_{n}$( u ) = 1 √$_{n}$ n ∑ i =1 ε$_{i}$ ( θ$_{0}$ )1 { q ( X$_{i}$, θ$_{0}$ ) ≤ u } − G $^{′}$( u, θ$_{0}$ $_{)}$√$_{n}$$_{(}$ ˆ θ$_{n}$ − θ$_{0}$ ) − G $^{′}$( u, θ$_{0}$ )Δ − $^{1}$( θ$_{0}$ ) [ 1 √$_{n}$ n ∑ i =1 ε$_{i}$ ( θ$_{0}$ ) g ( X$_{i}$, θ$_{0}$ ) − Δ( θ$_{0}$ $_{)}$√$_{n}$$_{(}$ ˆ θ$_{n}$ − θ$_{0}$ ) ] + o$_{p}$ (1) = 1 √$_{n}$ n ∑ i =1 ε$_{i}$ ( θ$_{0}$ ) ( 1 { q ( X$_{i}$, θ$_{0}$ ) ≤ u } − G $^{′}$( u, θ$_{0}$ )Δ − $^{1}$( θ$_{0}$ ) g ( X$_{i}$, θ$_{0}$ ) + o$_{p}$ (1)