[ − b, b ] for some 0 ≤ b < ∞ independent of P ; (S2) for all submodels dP$_{t}$/dP = 1 + th with bounded h with Ph = 0, there exists some δ > 0 and a set O$_{1}$ ⊆ O with P$_{0}$ ( O$_{1}$ ) = 1 such that, for all ( o, t$_{1}$ ) ∈ O$_{1}$ × ( − δ, δ ), t ↦→ T$_{P}$$_{t}$ ( x $^{T}$) is twice differentiable at t$_{1}$ with uniformly bounded (in x $^{T}$) first and second derivatives; (S3) for any P ∈ M and submodel dP$_{t}$/dP = 1+ th for bounded h with Ph = 0, there exists a function D T P : O →$_{R}$ uniformly bounded (in P and o ) such that ∫ D T $_{P}$( o ) dP ( o | x $^{T}$) = 0 for almost all o ∈ O and Condition (S1) ensures that T is bounded and only relies on a summary measure of an observation O . Condition (S2) ensures that we will be able to interchange differentiation and integration when needed. Condition (S3) is a conditional (and weaker) version of pathwise differentiability in that the typical inner product representation only needs to hold for the conditional distribution of O given X T under P$_{0}$ . We will verify in Section 5 that these conditions hold in the context of the motivating examples presented earlier. Remark 1. As a caution to the reader, we warn that simultaneously satisfying (S1) and (S3) may at times be restrictive. For example, if the observed data unit is O ≜ ( W (1) , W (2) , Y ) , the parameter cannot generally satisfy both conditions. In Section 5, we discuss this example further and provide a means to tackle this problem using the techniques we have developed. In concluding remarks, we discuss a weakening of our conditions, notably by replacing S by the linear span 8 T$_{P}$ ( o ) ≜ E$_{P}$ [ Y | W (1) = w (1) , W (2) = w (2)] − E$_{P}$ [ Y | W (1) = w (1)] d dt T$_{P}$$_{t}$ ( x $^{T}$)$^{∣}$ ∣ ∣ t =0 = ∫ D T $_{P}$( o ) h ( o ) dP ( o | x $^{T}$) .