Define the auxiliary empirical process Next lemma states that the unprojected process ˆ R$_{n}$ ( u ) defined in (2.10) is asymptotically equiv- alent under H$_{0}$ to the auxiliary process ˜ R$_{n}$ ( u ) defined above. Lemma A.3 Under Assumptions 3.1-3.3 and under the null hypothesis H$_{0}$ , we have Proof of Lemma A.3: Note that we have uniformly in u , where the second to last equality follows by the Taylor expansion of q ( X$_{i}$, ˆ θ$_{n}$ ) around q ( X$_{i}$, θ$_{0}$ ) and the Assumptions 3.1 and 3.2. Following the arguments in the proof of Theorem 1 in Stute and Zhu (2002), we next show that both A$_{1}$$_{n}$ ( u ) and A$_{2}$$_{n}$ ( u ) are uniformly negligible in u . For the first term A$_{1}$$_{n}$ ( u ), define the process Since under H$_{0}$ the ε$_{i}$ ’s are centered conditionally on X$_{i}$ ’s, α$_{n}$ ( u, θ ) has i.i.d. centered summands. Clearly, the first term A$_{1}$$_{n}$ ( u ) can be expressed as α$_{n}$ ( u, ˆ θ$_{n}$ ) − α$_{n}$ ( u, θ$_{0}$ ). From Lemma A.2, α$_{n}$ ( · , · ) is asymptotically equicontinuous, see e.g. Corollary 2.3.12 in van der Vaart and Wellner (1996). Since ˆ θ$_{n}$ → p θ$_{0}$ by Assumption 3.1, A$_{1}$$_{n}$ ( u ) → 0 in probability uniformly in u . For the second term A$_{2}$$_{n}$ ( u ), since by Assumption 3.1, √$_{n}$$_{(}$ ˆ θ$_{n}$ − θ$_{0}$ ) = O$_{p}$ (1), it remains to 35 ˜ R$_{n}$ ( u ) = 1 √$_{n}$ n ∑ i =1 ε$_{i}$ ( ˆ θ$_{n}$ )1 { q ( X$_{i}$, θ$_{0}$ ) ≤ u } . sup u ∈ Π ∣ ˆ R$_{n}$ ( u ) − ˜ R$_{n}$ ( u ) ∣ = o$_{p}$ (1) . ˆ R$_{n}$ ( u ) − ˜ R$_{n}$ ( u ) = 1 √$_{n}$ n ∑ i =1 ε$_{i}$ ( ˆ θ$_{n}$ ) ( 1 { q ( X$_{i}$, ˆ θ$_{n}$ ) ≤ u } − 1 { q ( X$_{i}$, θ$_{0}$ ) ≤ u } ) = 1 √$_{n}$ n ∑ i =1 ( ε$_{i}$ ( θ$_{0}$ ) − ( q ( X$_{i}$, ˆ θ$_{n}$ ) − q ( X$_{i}$, θ$_{0}$ ))) ( 1 { q ( X$_{i}$, ˆ θ$_{n}$ ) ≤ u } − 1 { q ( X$_{i}$, θ$_{0}$ ) ≤ u } ) = 1 √$_{n}$ n ∑ i =1 ε$_{i}$ ( θ$_{0}$ ) ( 1 { q ( X$_{i}$, ˆ θ$_{n}$ ) ≤ u } − 1 { q ( X$_{i}$, θ$_{0}$ ) ≤ u } ) − √$_{n}$$_{(}$ ˆ θ$_{n}$ − θ$_{0}$ ) $_{′}$1 n n ∑ i =1 g ( X$_{i}$, θ$_{0}$ ) ( 1 { q ( X$_{i}$, ˆ θ$_{n}$ ) ≤ u } − 1 { q ( X$_{i}$, θ$_{0}$ ) ≤ u } ) + o$_{p}$ (1) := A$_{1}$$_{n}$ ( u ) + A$_{2}$$_{n}$ ( u ) + o$_{p}$ (1) , α$_{n}$ ( u, θ ) = 1 √$_{n}$ n ∑ i =1 ε$_{i}$ ( θ$_{0}$ )1 { q ( X$_{i}$, θ ) ≤ u } .