4.1.1 Nuisance functions have been estimated well As we now establish, our test has power against all alternatives P$_{0}$ except for the fringe cases discussed in 1 with Γ$_{0}$ one-degenerate. We first note that When scaled by √ n , the leading term on the right-hand side follows a mean zero normal distribution under regularity conditions. The second summand is typically O$_{P}$$_{0}$ ( n − $^{1}$) under certain conditions, for example, on the entropy of the class of plausible realizations of the random function ( o$_{1}$, o$_{2}$ ) ↦→ Γ$_{n}$( o$_{1}$, o$_{2}$ ) (Nolan and Pollard, 1987, 1988). In view of the second statement in Theorem 3, the third summand is a second-order term that will often be negligible, even after scaling by √ n . As such, under certain regularity conditions, the leading term in the representation above determines the asymptotic behavior of ψ$_{n}$ , as described in the following theorem. Theorem 6 (Asymptotic distribution under H$_{1}$ ) . Suppose that K$_{1}$$_{n}$ = o$_{P}$$_{0}$ ( n − 1 / $^{2}$) , that $_{U$_{n}$}$˜ Γ$_{n}$ = o$_{P}$$_{0}$ ( n − 1 / $^{2}$) , and furthermore, that o ↦→ ∫ Γ$_{n}$( o$_{1}$, o ) dP$_{0}$ ( o ) belongs to a fixed P$_{0}$ - Donsker class with probability tending to 1 while ‖ P$_{0}$ (Γ$_{n}$ − Γ$_{0}$) ‖$_{2}$$_{,P}$$_{0}$ = o$_{P}$$_{0}$ (1) . If H$_{1}$ holds, we have that √ n ( ψ$_{n}$ − ψ$_{0}$ ) ⇝ N (0 , τ $^{2}$) , where τ 2 ≜ 4 V ar$_{P}$$_{0}$ [∫ Γ$_{0}$( O, o ) dP$_{0}$ ( o ) $^{]}$. In view of the results of Section 2, τ 2 coincides with σ 2 $_{0}$, the efficiency bound for regular, asymptotically linear estimators in a nonparametric model. Hence, ψ$_{n}$ is an asymptotically efficient estimator of ψ$_{0}$ under H$_{1}$ . Sufficient conditions for ∫ Γ$_{n}$( o$_{1}$, o ) dP$_{0}$ ( o ) to belong to a fixed P$_{0}$ -Donsker class with probability approaching one are given in Appendix B.2. The following corollary is trivial in light of Theorem 6. It establishes that the test nψ$_{n}$ > ˆ q ub 1 − α is consistent against (essentially) all alternatives provided the needed components of the likelihood are estimated sufficiently well. Corollary 3 (Consistency under a fixed alternative) . Suppose the conditions of Theorem 6. Furthermore, suppose that τ 2 > 0 and ˆ q ub 1 − α = o$_{P}$$_{0}$ ( n ) . Then, under H$_{1}$ , the test nψ$_{n}$ > ˆ q ub 1 − α 20 ψ$_{n}$ − ψ$_{0}$ = $_{U$_{n}$}$Γ$_{n}$ − ψ$_{0}$ = 2( P$_{n}$ − P$_{0}$ ) P$_{0}$ Γ$_{n}$ +$_{U}$$_{n}$ ˜ Γ$_{n}$ + Rem Ψ P .