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Abstract:
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We introduce a new method of estimation of parameters in semi- parametric and nonparametric models. The method is based U-statistics that are based on higher order influence functions that extend ordinary linear influence functions of the parameter of interest,and represent higher derivatives of this parameter. For parameters for which the representation cannot be perfect the method often leads to a bias-variance trade-off, and results in estimators that converge at slower than a root-n rate In a number of examples the resulting rate can be shown to be optimal. We are particularly interested in estimating parameters in models with a nuisance parameter of high dimension or low regularity, where the parameter of interest cannot be estimated at a root-n rate, but we also consider root-n efficient estimation using novel non-linear estimators. The general approach can be used to estimate a mean response when the response is missing at random, an average causal effect under ignorability, and to construct adaptive confidence intervals for a covariate- specific treatment effect function in randomized trials.
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