Note that the integrating measure in CvM$_{n}$ is a random measure, but Corollary 1 shows that the asymptotic distribution is not affected by this fact. Further details can be found in the Appendix A. Now, we investigate the power properties of tests based on continuous functionals Γ( ˆ R p $_{n}$), like CvM$_{n}$ and KS$_{n}$ in (2.8) and (2.9), respectively. We consider fixed alternatives, and a sequence of local alternatives H$_{1}$$_{n}$ that converges to H$_{0}$ at the parametric rate n − 1 / $^{2}$. 3.2 Asymptotic power 3.2.1 Power against fixed alternatives Next theorem analyzes the asymptotic properties of our tests under fixed alternatives of the type where Π = [0 , 1] is the unit interval. Note that H$_{1}$ is simply the negation of H$_{0}$ in (2.3). Theorem 2 Suppose Assumptions 3.1-3.3 hold. Then, under the fixed alternative hypothesis H$_{1}$ in (3.2), we have that H$_{1}$ : E [( D − q ( X, θ ))1 { q ( X, θ ) ≤ u } ] = 0 for all θ ∈ Θ and for some u ∈ Π , (3.2) From Theorem 2, we see that test statistics of the form of Γ( ˆ R p $_{n}$) are not consistent against all fixed alternative hypotheses in (3.2), but only those not collinear to the score function g ( X, θ$_{0}$ ). To see this, note that is equal to zero under (3.2) if p ( X ) − q ( X, θ$_{0}$ ) and g ( X, θ$_{0}$ ) are collinear almost surely. We do not see this as a limitation. First, when one estimates θ$_{0}$ using the NLS method, the population first order condition for θ$_{0}$ sets E [( D − q ( X, θ$_{0}$ )) g $^{′}$( X, θ$_{0}$ )] = 0 , implying that, for some u ∈ Π , E [( p ( X ) − q ( X, θ$_{0}$ )) P 1 { q ( X, θ$_{0}$ ) ≤ u } ] = E [( p ( X ) − q ( X, θ$_{0}$ )) 1 { q ( X, θ$_{0}$ ) ≤ u } ] − E [ ( p ( X ) − q ( X, θ$_{0}$ )) g $^{′}$( X, θ$_{0}$ ) ] Δ − $^{1}$( θ$_{0}$ ) G ( u, θ$_{0}$ ) E [( p ( X ) − q ( X, θ$_{0}$ )) P 1 { q ( X, θ$_{0}$ ) ≤ u } ] = E [( p ( X ) − q ( X, θ$_{0}$ )) 1 { q ( X, θ$_{0}$ ) ≤ u } ] = 0 . 13 sup u ∈ Π ∣ 1 √$_{n}$ ˆ R p $_{n}$( u ) − E [( p ( X ) − q ( X, θ$_{0}$ )) P 1 { q ( X, θ$_{0}$ ) ≤ u } ] ∣ = o$_{p}$ (1) .