that Γ$_{P}$ ≜ Γ RR P − 2Γ RS P + Γ SS P yields the result. Proof of Theorem 3. Fix P ∈ M and let P$_{0}$ satisfy H$_{0}$ . We have that P 2 $_{0}$Γ$_{P}$ − ψ$_{0}$ = P 2 $_{0}$Γ RR P − Φ $^{RR}$( P$_{0}$ ) + P 2 $_{0}$Γ SS P − Φ $^{SS}$( P$_{0}$ ) − $^{[}$P 2 $_{0}$Γ RS P − Φ $^{RS}$( P$_{0}$ ) + P 2 $_{0}$Γ SR P − Φ $^{SR}$( P$_{0}$ ) $^{]}$. Taking the absolute value of both sides, applying the triangle inequality, and using A.5 yields ∣ $_{∣}$P 2 $_{0}$Γ$_{P}$ − ψ$_{0}$ ∣ ≲ Rem Φ RR P + Rem Φ SS P +2 Rem Φ RS P ≲ ‖ L RS P ‖ 2 1 ,P$_{0}$ + ‖ M RS P ‖ 4 4 ,P$_{0}$ + ‖ L RS P ‖$_{2}$$_{,P}$$_{0}$ ‖ M RS P ‖$_{2}$$_{,P}$$_{0}$ , where the final inequality uses the maximum in the definition of L RS P and M RS P . The inequality for when P$_{0}$ satisfies H$_{1}$ is proven in A.6. Proof of 1. By the first result of A.6, | P$_{0}$ Γ$_{n}$( o$_{1}$, · ) | ≲ K$_{n}$ for P$_{0}$ almost all o$_{1}$ ∈ O $^{′}$. We have that The fact that $^{{}$o$_{1}$ ↦→ P$_{0}$ Γ$_{n}$( o$_{1}$, · ) K$_{n}$ : ˆ P$_{n}$ } belongs to a P$_{0}$ Donsker class with probability ap- proaching 1, where ˆ P$_{n}$ varies over the set of its possible realizations, yields that ( P$_{n}$ − P$_{0}$ ) ( P$_{0}$ Γ$_{n}$ K$_{n}$ ) = O$_{P}$$_{0}$ ( n − 1 / $^{2}$) (van der Vaart and Wellner, 1996), and thus the right-hand side above is O$_{P}$$_{0}$ ( K$_{n}$/ √ n ). If K$_{n}$ = o$_{P}$$_{0}$ ( n − 1 / $^{2}$), then this yields that the right-hand side above is o$_{P}$$_{0}$ ( n − $^{1}$). Proof of Theorem 4. Plugging C1), C2), and C3) into (4) yields 53 ψ$_{n}$ − ψ$_{0}$ =$_{U}$$_{n}$Γ$_{0}$ + o$_{P}$$_{0}$ ( n − $^{1}$) . (A.12) | ( P$_{n}$ − P$_{0}$ ) P$_{0}$ Γ$_{n}$ | = K$_{n}$$^{∣}$ ∣ ∣ ( P$_{n}$ − P$_{0}$ )$^{(}$ P$_{0}$ Γ$_{n}$ K$_{n}$ ) ∣ ∣ . 7.1 Proofs for Section 3