by ULLN of Newey and McFadden (1994) and similar arguments in proving Lemmas A.3, A.6 and A.7. □ Proof of Theorem 3: Note that under the local alternatives H$_{1}$$_{n}$ in (3.4), we have that uniformly in u ∈ Π: ˆ R p $_{n}$( u ) = 1 √$_{n}$ n ∑ i =1 ( ε$_{i}$ ( ˆ θ$_{n}$ ) − r ( q ( X$_{i}$, ˆ θ$_{n}$ )) √$_{n}$ ) P $_{n}$1 { q ( X$_{i}$, ˆ θ$_{n}$ ) ≤ u } + 1 n n ∑ i =1 r ( q ( X$_{i}$, ˆ θ$_{n}$ )) P $_{n}$1 { q ( X$_{i}$, ˆ θ$_{n}$ ) ≤ u } = 1 √$_{n}$ n ∑ i =1 ( ε$_{i}$ ( θ$_{0}$ ) − r ( q ( X$_{i}$, θ$_{0}$ )) √$_{n}$ ) P 1 { q ( X$_{i}$, θ$_{0}$ ) ≤ u } + E [ r ( q ( X, θ$_{0}$ )) P 1 { q ( X, θ$_{0}$ ) ≤ u } ] + o$_{p}$ (1) := R p n $_{1}$( u ) + Δ$_{r}$( u ) + o$_{p}$ (1) ⇒ R p ∞ + Δ$_{r}$ , where the second equality follows by similar arguments in proving Theorem 1 and by ULLN. Since ε$_{i}$ ( θ$_{0}$ ) − n − 1 / $^{2}$r ( q ( X$_{i}$, θ$_{0}$ )) forms a zero mean and i.i.d. summand in this local alternative framework, we can apply the functional central limit theorem to R p n $_{1}$( u ), just as we applied it to R p n $_{0}$( u ) defined in (2.14), leading to R p n $_{1}$( u ) ⇒ R p $_{∞}$. The last step then follows and we finish the proof of Theorem 3. □ Proof of Theorem 4: As in Theorem 1, we have the following decomposition: By Lemma A.2, it follows from a stochastic equicontinuity argument and the consistency of ˆ θ$_{n}$ to θ$_{0}$ that, uniformly in u ∈ Π, 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$_{n}$ ) V$_{i}$ = 1 √$_{n}$ n ∑ i =1 ε$_{i}$ ( ˆ θ$_{n}$ )1 { q ( X$_{i}$, ˆ θ$_{n}$ ) ≤ u } V$_{i}$ − 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$_{n}$ ) V$_{i}$ := ˆ R ∗ $_{n}$( u ) − G ′ $_{n}$( u, ˆ θ$_{n}$ )Δ − 1 n ( ˆ θ$_{n}$ ) ˆ S ∗ $_{n}$. ˆ R ∗ $_{n}$( u ) = 1 √$_{n}$ n ∑ i =1 ε$_{i}$ ( θ$_{0}$ )1 { q ( X$_{i}$, θ$_{0}$ ) ≤ u } V$_{i}$ + o$_{p}$ (1) , 40 = sup u ∈ Π ∣ 1 √$_{n}$ ˆ R p $_{n}$( u ) − E [( p ( X ) − q ( X, θ$_{0}$ )) P 1 { q ( X, θ$_{0}$ ) ≤ u } ] ∣ = o$_{p}$ (1)