Robins and Rotnitzky (2004) has established that U is an influence function for γ in the semiparametric model where assumptions (1)-(4) hold. This, in turn, implies that

E [∂U(ψ0, η, α, ω)

∂(η, α, ω) ∣(η,α,ω)=(η0,α0,ω0)] = 0.

Hence, under the null hypothesis (H0) i.e. that of no modelling error for the nuisance

models, ∑ i ̂ Ui≈ ∑ i {Ui+ E [ ∂U(ψ) ∂ψ ∣ψ=ψ0]E [S eff ψS effT ψ ] −1 Seff ψ,i} , where S eff

ψ is the efficient score

of ψ under our parametric model. By a property of influence functions,

E [∂U(ψ) ∂ψ ∣ψ=ψ0] = −E [US effT ψ ] . Thus, n−1/2∑ i ̂ U ≈ n−1/2∑ i {Ui− E [US effT ψ ] E [S eff ψ S effT ψ ] −1 Seff

ψ,i} and we have

E[{Ui− E [US effT ψ ] E [S eff ψS effT ψ ] −1 Seff ψ,i} 2 ] ≤ E {U2 i} (A.1)

since E [USeffT ψ ] E [S eff ψS effT ψ ] −1 Seff

ψ,i is the orthogonal projection of U onto the span of S

eff

ψ,i,

proving the first result. The second result follows from a standard application of Slutsky’s theorem and the central limit theorem.

Acknowledgments: The authors acknowledge research support from the National In- stitutes of Health (NIH). R. A. Matsouaka was supported by NIH grants 1R01MD006064, 1R21HD066312 (Theresa Osypuk, PI) and 1R21ES019712. E. Tchetgen Tchetgen was sup- ported by NIH grants 1R21ES019712, R01AI104459 and R01HL080644. We are also grateful to Nicole Schmidt for helpful comments and suggestions regarding the MTO data.

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