of elements in S . Consideration of this larger class significantly complicates the form of the estimator we propose in Section 3. We are now in a position to discuss the pathwise differentiability of Ψ. For any elements T, U ∈ S , we define and set Γ$_{P}$ ≜ Γ RR P − Γ RS P − Γ SR P + Γ SS P . Note that Γ$_{P}$ is symmetric for any P ∈ M . For brevity, we will write Γ TU 0 and Γ$_{0}$ to denote Γ TU P$_{0}$ and Γ$_{P}$$_{0}$, respectively. The following theorem characterizes the first-order behavior of Ψ at an arbitrary P ∈ M . Theorem 1 (First-order pathwise differentiability of Ψ over M ) . If R, S ∈ S , the parameter Ψ : M →$_{R}$ is pathwise differentiable at P ∈ M with first-order canonical gradient given by D Ψ 1 ( P )( o ) ≜ 2 [∫ Γ$_{P}$ ( o, o$_{2}$ ) dP ( o$_{2}$ ) − Ψ( P ) $^{]}$. Under some conditions, it is straightforward to construct an asymptotically linear esti- mator of ψ$_{0}$ with influence function D Ψ 1 ( P$_{0}$ ), that is, an estimator ψ$_{n}$ of ψ$_{0}$ such that For example, the one-step Newton-Raphson bias correction procedure (see, e.g., Pfanzagl, 1982) or targeted minimum loss-based estimation (see, e.g., van der Laan and Rose, 2011) can be used for this purpose. If the above representation holds and the variance of D Ψ 1 ( P$_{0}$ )( O ) is positive, then √ n ( ψ$_{n}$ − ψ$_{0}$ ) ⇝ N (0 , σ 2 $_{0}$), where the symbol ⇝ denotes convergence in distribution and we write σ 2 0 ≜ P$_{0}$ $^{[}$D Ψ 1 ( P$_{0}$ ) 2 $^{]}$. If σ$_{0}$ is strictly positive and can be consistently estimated, Wald-type confidence intervals for ψ$_{0}$ with appropriate asymptotic coverage can be constructed. 9 ψ$_{n}$ − ψ$_{0}$ = 1 n n ∑ i =1 D Ψ 1 ( P$_{0}$ )( O$_{i}$ ) + o$_{P}$$_{0}$ ( n − 1 / $^{2}$) . Γ TU P ( o$_{1}$, o$_{2}$ ) ≜$^{[}$ 2 [ T$_{P}$ ( o$_{1}$ ) − U$_{P}$ ( o$_{2}$ )] $^{[}$D U $_{P}$( o$_{2}$ ) − D T $_{P}$( o$_{1}$ ) ] + 1 − $^{{}$4 [ T$_{P}$ ( o$_{1}$ ) − U$_{P}$ ( o$_{2}$ )] 2 − 2 $^{}}$D T $_{P}$( o$_{1}$ ) D U $_{P}$( o$_{2}$ )$^{]}$ e − [ T$_{P}$ ( o$_{1}$ ) − U$_{P}$ ( o$_{2}$ )] 2 .