Abstract:
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We consider estimation of the total effect of time dependent treatments on an outcome assuming a causal graphical model with hidden variables. The ID algorithm (Tian and Pearl, 2002) determines if the effect is identified and when it is, it returns a functional ?(P) of the observed data law P that identifies it. Because ?(P) typically depends on infinite dimensional parameters, such as conditional means or densities, plug-in estimators ?(P_{n}) do not converge at rate root-n for most non-parametric estimators P_{n} of P. One-step cross-fitted estimation is a well known strategy for correcting the bias of ?(P_{n}) and hopefully yielding root-n consistent estimation. It is based on adding to ?(P_{n}) the empirical mean of an estimator of the influence function of ?(P) with unknown nuisance functionals ?(P) estimated by plug-in from an independent sample. In this talk we argue that we can significantly improve upon one-step cross-fitted estimators that depend on plug-in estimators of ?(P) by considering estimators of ?(P) that minimize an objective function which is itself a debiased estimator of a population objective function that is minimized at ?(P)
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