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M. van der Laan 475has the general property thatP 0 D ∗ (Q, g) =Ψ(Q 0 ) − Ψ(Q)+R(Q, Q 0 ,g,g 0 )for a second order term R(Q, Q 0 ,g,g 0 ) . In fact, in many applications, we havethat R(Q, Q 0 ,g,g 0 ) equals an integral of (Q − Q 0 )(g − g 0 ) so that it equalszero if either Q = Q 0 or g = g 0 , which is often referred to as double robustnessof the efficient influence curve. In that case, P n D ∗ (Q ∗ n,g 0 )=0impliesthatΨ(Q ∗ n)isaconsistentestimatorofψ 0 . In essence, the norm of P n D ∗ (Q ∗ n,g 0 )represents a criterion measuring a distance between Ψ(Q ∗ n) and ψ 0 , so thatminimizing the Euclidean norm of P n D ∗ (Q ∗ n,g n )correspondswithfittingψ 0 .Since in many applications, the nuisance parameter g 0 is unknown, one willhave to replace g 0 in the updating procedure by an estimator g n .Inthatcase,we haveP 0 D ∗ (Q ∗ n,g n )=ψ 0 − Ψ(Q ∗ n)+R(Q ∗ n,Q 0 ,g n ,g 0 ),where the remainder is still a second order term but now also involving crosstermdifferences (Q ∗ n − Q 0 )(g n − g 0 ).40.5.2 Asymptotic linearity of TMLEIf this second order remainder term R(Q ∗ n,Q 0 ,g n ,g 0 )convergestozeroinprobability at a rate faster than 1/ √ n,thenitfollowsthatψ ∗ n − ψ 0 =(P n − P 0 )D ∗ (Q ∗ n,g n )+o P (1/ √ n),so that, if P 0 {D ∗ (Q ∗ n,g n ) − D ∗ (Q 0 ,g 0 )} 2 → 0inprobability,andtherandomfunction D ∗ (Q ∗ n,g n ) of O falls in a P 0 -Donsker class, it follows thatψ ∗ n − ψ 0 =(P n − P 0 )D ∗ (Q 0 ,g 0 )+o P (1/ √ n).That is, √ n (ψ ∗ n − ψ 0 ) is asymptotically Normally distributed with mean zeroand variance equal to the variance of the efficient influence curve. Thus, ifQ ∗ n,g n are consistent at fast enough rates, then ψ ∗ n is asymptotically efficient.Statistical inference can now be based on the Normal limit distribution andan estimator of its asymptotic variance, such as σ 2 n = P n D ∗ (Q ∗ n,g n ) 2 .Thisdemonstrates that the utilization of the state of the art in adaptive estimationwas not a hurdle for statistical inference, but, on the contrary, it is required toestablish the desired asymptotic Normality of the TMLE. Establishing asymptoticlinearity of TMLE under misspecification of Q 0 (in the context that theefficient influence curve is double robust), while still allowing the utilization ofvery adaptive estimators of g 0 ,hastodealwithadditionalchallengesresolvedby also targeting the fit of g; see,e.g.,vanderLaan(2012)andvanderLaanand Rose (2012).