is consistent in the sense that The requirement that ˆ q ub 1 − α = o$_{P}$$_{0}$ ( n ) is very mild given that q$_{1}$$_{−}$$_{α}$ will be finite whenever R, S ∈ S . As such, we would not expect ˆ q ub 1 − α to get arbitrarily large as sample size grows, at least beyond the extent allowed by our corollary. This suggests that most non-trivial upper bounds satisfying (6) will yield a consistent test. We now consider the case where the nuisance functions are not estimated well, in the sense that the consistency conditions of Theorem 6 do not hold. In particular, we argue that failure of these conditions does not necessarily undermine the consistency of our test. Let ˆ q ub 1 − α be the estimated cutoff for our test, and suppose that ˆ q ub 1 − α = o$_{P}$$_{0}$ ( n ). Suppose also that P 2 $_{0}$Γ$_{n}$ is asymptotically bounded away from zero in the sense that, for some δ > 0, P n 0 ( P 2 $_{0}$Γ$_{n}$ > δ ) tends to one. This condition is reasonable given that P 2 $_{0}$Γ$_{0}$ > 0 if H$_{1}$ holds and ˆ P$_{n}$ is nevertheless a (possibly inconsistent) estimator of P$_{0}$ . Assuming that [$_{U}$$_{n}$ − P 2 $_{0}$]Γ$_{n}$ = O$_{P}$$_{0}$ ( n − 1 / $^{2}$), which is true under entropy conditions on Γ$_{n}$ (Nolan and Pollard, 1987, 1988), we have that We have accounted for the random n − 1 / $^{2}$ˆ q ub 1 − α term as in the proof of 3. Of course, this result is less satisfying than Theorem 6, which provides a concrete limit distribution. 21 P n 0 $^{(}$nψ$_{n}$ > ˆ q ub 1 − α ) = P n 0 ( √ n [$_{U}$$_{n}$ − P 2 $_{0}$]Γ$_{n}$ > ˆ q ub 1 − α √ n − √ nP 2 $_{0}$Γ$_{n}$$^{)}$ −→ 1 . 4.1.2 Nuisance functions have not been estimated well lim n $_{→∞}$P n 0 $^{(}$nψ$_{n}$ > ˆ q ub 1 − α ) = 1 .