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M. van der Laan 46940.2.3 Target quantity of interest, and its identifiability fromobserved data distributionThe importance of constructing a model is that it allows us to define interestingtarget quantities Ψ F (θ) for a given mapping Ψ F :Θ→ IR d that representthe scientific question of interest that we would like to learn from our dataO 1 ,...,O n .GivensuchadefinitionoftargetparameterΨ F :Θ→ IR d one likesto establish that there exists a mapping Ψ : M→IR d so that Ψ F (θ) =Ψ(P θ ).If such a mapping Ψ exists we state that the target quantity Ψ F (θ) isidentifiablefrom the observed data distribution. This is often only possible bymaking additional non-testable restrictions on θ in the sense that one is onlyable to write Ψ F (θ) =Ψ(P θ ) for θ ∈ Θ ∗ ⊂ Θ.40.2.4 Statistical target parameter/estimand, and thecorresponding statistical estimation problemThis identifiability result defines a statistical target parameter Ψ : M →IR d .Thegoalistoestimateψ 0 =Ψ(P 0 )basedonn i.i.d. observations onO ∼ P 0 ∈M.Theestimandψ 0 can be interpreted as the target quantityΨ F (θ 0 ) if both the non-testable and the statistical model assumptions hold.Nonetheless, due to the statistical model containing the true data distributionP 0 , ψ 0 always has a pure statistical interpretation as the feature Ψ(P 0 ) of thedata distribution P 0 . A related additional goal is to obtain a confidence intervalfor ψ 0 . A sensitivity analysis can be used to provide statistical inference for theunderlying target quantity ψ0F under a variety of violations of the assumptionsthat were needed to state that ψ0F = ψ 0 , as we will discuss shortly below.40.3 The curse of dimensionality for the MLE40.3.1 Asymptotically linear estimators and influence curvesAn estimator is a Euclidean valued mapping ˆΨ on a statistical model thatcontains all empirical probability distributions. Therefore, one might representan estimator as a mapping ˆΨ :M NP → IR d from the nonparametricstatistical model M NP into the parameter space. In order to allow for statisticalinference, one is particularly interested in estimators that behave in firstorder as an empirical mean of i.i.d. random variables so that it is asymptoticallyNormally distributed. An estimator ˆΨ is asymptotically linear at datadistribution P 0 with influence curve D 0 ifˆΨ(P n ) − ψ 0 = 1 nn∑D 0 (O i )+o P (1/ √ n).i=1