ω (0) = 0 such that Remark 8. A sufficient, but not necessary, condition for such an ω to exist is that F$_{π}$ is twice continuously differentiable with the absolute range of the second derivative ¨ F$_{π}$ on [ V ( π ) ± c ] bounded away from infinity uniformly in π ∈ Π. Indeed, Taylor’s theorem then shows that one can take The next lemma gives conditions under which (9) holds. Its proof is given in Ap- pendix B.3. Lemma 9. If (15) and (16) are met, then (9) holds. For every π ∈ P , V ( π ) defined in (14) can be viewed as the evaluation at P of the functional from the nonparametric model of distributions P ′ satisfying the same constraints as P to the real line. This functional is pathwise differentiable at P relative to the maximal tangent space with an efficient influence function f$_{π}$ given by Observe that above ˙ F$_{π}$ ( V ( π )) only enters f$_{π}$ as a multiplicative constant, and therefore an estimating equation-based estimator or TMLE for V ( π ) can be asymptotically linear for V ( π ) 16 sup π ∈ Π sup | k |≤ c [∣ ∣ ˙ F$_{π}$ ( V ( π ) + k ) − ˙ F$_{π}$ ( V ( π )) ∣ − ω ( | k | ) ] ≤ 0 . (16) ω ( m ) ≡ m × sup π ∈ Π sup ˜ m ∈ [ V ( π ) ± c ] | ¨ F$_{π}$ ( ˜ m ) | . P ′ ↦→ inf { m ∈ R : 1 / 2 ≤ E P ′ [ P $^{′}$( Y ≤ m | A = π ( X ) , X )] } f$_{π}$ ( o ) ≡ 1 { a = π ( x ) } P ( A = a | X = x ) ˙ F$_{π}$ ( V ( π )) [ 1 { y ≤ V ( π ) } − P { Y ≤ V ( π ) | A = a, X = x } ] + P { Y ≤ V ( π ) | A = π ( x ) , X = x } − 1 / 2 ˙ F$_{π}$ ( V ( π )) . (17)