4.2 Consistency under a local alternative We consider local alternatives of the form where h$_{n}$ → h in L 2 $_{0}$( P$_{0}$ ) for some non-degenerate h and P$_{0}$ satisfies the null hypothesis H$_{0}$ . Suppose that the conditions of Theorem 4 hold. By Theorem 2.1 of Gregory (1977), we have that where$_{U}$$_{n}$ is the U -statistic empirical measure from a sample of size n drawn from Q$_{n}$ , 〈· , ·〉 is the inner product in L $^{2}$( P$_{0}$ ), Z$_{k}$ and λ$_{k}$ are as in Theorem 4, and f$_{k}$ is the eigenfunction corresponding to eigenvalue λ$_{k}$ described in Theorem 4. By the contiguity of Q$_{n}$ , the condi- tions of Theorem 4 yield that the result above also holds with$_{U}$$_{n}$Γ$_{0}$ replaced by$_{U}$$_{n}$Γ$_{n}$, our estimator applied to a sample of size n drawn from Q$_{n}$ . If each λ$_{k}$ is non-negative, the limiting distribution under Q$_{n}$ stochastically dominates the asymptotic distribution under P$_{0}$ , and furthermore, if 〈 f$_{k}$, h 〉 = 0 for some k with λ$_{k}$ > 0, this dominance is strict. It is straightforward to show that, under the conditions of Theorem 4, the above holds if and only if lim inf$_{n}$ √ n Ψ( Q$_{n}$ ) > 0, that is, if the sequence of alternatives is not too hard. Suppose that ˆ q$_{1}$$_{−}$$_{α}$ is a consistent estimate of q$_{1}$$_{−}$$_{α}$ . By Le Cam’s third lemma, ˆ q$_{1}$$_{−}$$_{α}$ is consistent for q$_{1}$$_{−}$$_{α}$ even when the estimator is computed on samples of size n drawn from Q$_{n}$ rather than P$_{0}$ . This proves the following theorem. Theorem 7 (Consistency under a local alternative) . Suppose that the conditions of Theo- rem 4 hold. Then, under H$_{0}$ and provided lim inf$_{n}$$_{→∞}$ √ n Ψ( Q$_{n}$ ) > 0 , the proposed test is locally consistent in the sense that lim$_{n}$$_{→∞}$ Q$_{n}$ ( nψ$_{n}$ > ˆ q$_{1}$$_{−}$$_{α}$ ) > α , where ˆ q$_{1}$$_{−}$$_{α}$ is a consistent estimator of q$_{1}$$_{−}$$_{α}$ . 22 n$_{U}$$_{n}$ Γ$_{0}$ Q$_{n}$ ⇝ ∞ ∑ k =1 λ$_{k}$ $^{[}$( Z$_{k}$ + 〈 f$_{k}$, h 〉 ) 2 − 1 $^{]}$, dQ$_{n}$ ( o ) = $^{[}$1 + n − 1 / $^{2}$h$_{n}$ ( o ) $^{]}$dP$_{0}$ ( o ) ,