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The situation is more challenging if σ$_{0}$ = 0. In this case, √ n ( ψ$_{n}$ − ψ$_{0}$ ) → 0 in probability and typical Wald-type confidence intervals will not be appropriate. Because D Ψ 1 ( P$_{0}$ )( O ) has mean zero under P$_{0}$ , this happens if and only if D Ψ 1 ( P$_{0}$ ) ≡ 0. The following lemma provides necessary and sufficient conditions under which σ$_{0}$ = 0. 

Corollary 1 (First-order degeneracy under H$_{0}$ ) . If R,S ∈ S , it will be the case that σ$_{0}$ = 0 if and only if either (i) H$_{0}$ holds, or (ii) R$_{0}$ ( O ) and S$_{0}$ ( O ) are degenerate, i.e. almost surely constant but not necessarily equal, with D R 0 ≡ D S $_{0}$. 

The above results rely in part on knowledge of D R 0 and D S $_{0}$. It is useful to note that, in some situations, the computation of D T $_{P}$( o ) for a given T ∈ S and P ∈ M can be streamlined. This is the case, for example, if P ↦→ T$_{P}$ is invariant to fluctuations of the marginal distribu- tion of X $^{T}$, as it seems (S3) may suggest. Consider obtaining iid samples of increasing size from the conditional distribution of O given X T = x T under P , so that all individuals have observed X T = x $^{T}$. Consider the fluctuation submodel dP$_{t}$ ( o | x $^{T}$) ≜ [1 + th ( o )] dP ( o | x $^{T}$) for the conditional distribution, where h is uniformly bounded and ∫ h ( o ) dP ( o | x $^{T}$) = 0. Sup- pose that (i) P ↦→ T$_{P}$ ( x $^{T}$) is differentiable at t = 0 with respect to the above submodel and (ii) this derivative satisfies the inner product representation 

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for some uniformly bounded function o ↦→ ˜ D T $_{P}$( o | x $^{T}$) with ∫ ˜ D T $_{P}$( o | x $^{T}$) dP ( o | x $^{T}$) = 0. If the above holds for all x $^{T}$, we may take D T $_{P}$( o ) = ˜ D T $_{P}$( o | x $^{T}$) for all o with X $^{T}$( o ) = x $^{T}$. If D T P is uniformly bounded in P , (S3) then holds. 

In summary, the above discussion suggests that, if T is invariant to fluctuations of the marginal distribution of X $^{T}$, (S3) can be expected to hold if there exists a regular, asymptot- ically linear estimator of each T$_{P}$ ( x $^{T}$) under iid sampling from the conditional distribution of O given X T = x T implied by P . 

Remark 2. If T is invariant to fluctuations of the marginal distribution of X $^{T}$, one can 

