Proof of Theorem 6. We have By assumption, $_{U$_{n}$}$˜ Γ$_{n}$ = o$_{P}$$_{0}$ ( n − 1 / $^{2}$). The final term is o$_{P}$$_{0}$ ( n − 1 / $^{2}$) by the Donsker condition and the consistency condition (van der Vaart and Wellner, 1996). By the second result of A.6 and the assumption that K$_{n}$ = o$_{P}$$_{0}$ ( n − 1 / $^{2}$), this yields that Multiplying both sides by √ n , and applying the central limit theorem yields the result. Proof of 3. We have that Fix 0 < ϵ < ψ$_{0}$ . The right-hand side is equal to where Z ∼ N (0 , 1). The final equality holds by Theorem 6 and the well known result about the uniform convergence of distribution functions at continuity points when random variables converge in distribution (see, e.g., Theorem 5.6 in Boos and Stefanski, 2013). The result follows by noting that ( ϵ − ψ$_{0}$ ) /σ$_{0}$ is negative and that lim$_{z}$$_{→−∞}$ P r ( Z ≤ z ) = 0. 55 P n 0 $_{{}$√ n ( ψ$_{n}$ − ψ$_{0}$ ) σ$_{0}$ ≤ ˆ q ub 1 − $_{α}$n − 1 / 2 − √ nψ$_{0}$ σ$_{0}$ and ˆ q ub 1 − $_{α}$n − 1 ≤ ϵ$^{}}$ + o (1) ≤ P n 0 $_{{}$√ n ( ψ$_{n}$ − ψ$_{0}$ ) σ$_{0}$ ≤ √ n ( ϵ − ψ$_{0}$ ) σ$_{0}$ and ˆ q ub 1 − $_{α}$n − 1 ≤ ϵ$^{}}$ + o (1) ≤ P n 0 $_{{}$√ n ( ψ$_{n}$ − ψ$_{0}$ ) σ$_{0}$ ≤ √ n ( ϵ − ψ$_{0}$ ) σ$_{0}$ } + o (1) = P r$^{{}$Z ≤ √ n ( ϵ − ψ$_{0}$ ) σ$_{0}$ } + o (1) , P n 0 $^{{}$nψ$_{n}$ ≤ ˆ q ub 1 − α $^{}}$= P n 0 $_{{}$√ n ( ψ$_{n}$ − ψ$_{0}$ ) σ$_{0}$ ≤ ˆ q ub 1 − $_{α}$n − 1 / 2 − √ nψ$_{0}$ σ$_{0}$ } ψ$_{n}$ − ψ$_{0}$ = 2( P$_{n}$ − P$_{0}$ ) P$_{0}$ Γ$_{0}$ + o$_{P}$$_{0}$ ( n − 1 / $^{2}$) . ψ$_{n}$ = 2( P$_{n}$ − P$_{0}$ ) P$_{0}$ Γ$_{n}$ + P 2 $_{0}$Γ$_{n}$ +$_{U}$$_{n}$ ˜ Γ$_{n}$ = 2( P$_{n}$ − P$_{0}$ ) P$_{0}$ Γ$_{0}$ + P 2 $_{0}$Γ$_{n}$ +$_{U}$$_{n}$ ˜ Γ$_{n}$ + 2( P$_{n}$ − P$_{0}$ ) P$_{0}$ (Γ$_{n}$ − Γ$_{0}$) .