of view, the benefits of using our procedure can outweigh the costs in many relevant situations. Next, we study the performance of our projection-based tests under a sequence of local alter- native hypotheses converging to the null at the parametric rate n − 1 / 2 given by 3.2.2 Power against local alternatives for some θ$_{0}$ ∈ Θ, where r ( q ( X, θ$_{0}$ )) represents directions of departure from H$_{0}$ , and n − 1 / 2 indicates the rate of convergence of H$_{1}$$_{n}$ to H$_{0}$ . The function r : [0 , 1] → R is required to satisfy the following assumption. Assumption 3.4 The function r ( q ) is continuous in q and satisfies $^{E}$| r ( q ( X, θ$_{0}$ )) | < ∞ . Theorem 3 Suppose Assumptions 3.1 -3.4 hold. Then, under the local alternatives H$_{1}$$_{n}$ given by (3.4), we have where R p ∞ is the same Gaussian process as defined in Theorem 1, and Δ$_{r}$ is a deterministic shift function given by Note that, in general, the deterministic shift function Δ$_{r}$ ( u ) = 0 for at least some u ∈ Π, implying that tests based on continuous even functionals of ˆ R p $_{n}$( · ) will have non-trivial power against local alternatives of the form in (3.4). A situation in which our tests will have trivial local power against such alternatives is when directions r ( q ( x, θ$_{0}$ )) are a linear combination of score function g ( x, θ$_{0}$ ), i.e. r ( q ( x, θ$_{0}$ )) = βg ( x, θ$_{0}$ ) for some β . In such a case, the limiting distribution of ˆ R p $_{n}$( u ) under H$_{0}$ and H$_{1}$$_{n}$ is the same so that H$_{1}$$_{n}$ cannot be detected. On the other hand, note that tests based on the local smoothing approach such as Shaikh et al. (2009) are not able to detect alternatives of the form (3.4). From the above theorems, we see that the asymptotic distribution of continuous functionals Γ ( ˆ R p n ) depend on the underlying data generating process and of course on Γ ( · ) itself. Fur- thermore, the complicated covariance structure of K $^{p}$( · , · ) given in (3.1) does not allow for a 3.3 Computation of critical values 16 Δ$_{r}$( u ) ≡ E [ r ( q ( X, θ$_{0}$ )) P 1 { q ( X, θ$_{0}$ ) ≤ u } ] . ˆ R p $_{n}$( u ) ⇒ R p ∞ + Δ$_{r}$ , H$_{1}$$_{n}$ : E [ D − q ( X, θ$_{0}$ ) | q ( X, θ$_{0}$ )] = r ( q ( X, θ$_{0}$ )) √$_{n}$ a.s. (3.4)