Characterization of parameters with a mixed bias property
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In this article we characterize a class of parameters in large non-parametric models that admit rate doubly robust estimators. An estimator of a parameter of interest which relies on non-parametric estimators of two nuisance functions is rate doubly robust if it is consistent and asymptotically normal when one succeeds in estimating both nuisance functions at sufficiently fast rates, with the possibility of trading off slower rates of convergence for estimators of one of the nuisance functions with faster rates of convergence of the estimator of the other nuisance function. Our class is defined by the property that the bias of the one step estimator of the parameter of interest is the mean of the product of the estimation errors of the two nuisance functions. We call this property the mixed bias property. We show that our class strictly includes two recently studied classes of parameters that satisfy the mixed biased property and which include many important parameters of interest in causal inference. For parameters in our class we characterize their form and the form of their influence functions. Furthermore, we derive two functional moment equations, each being solved at one of the two nuisance functions. In addition, we derive two functional loss functions, each having expectation that is minimized at one of the two nuisance functions. Both the moment equations and the loss functions are important because they can be used to derive loss based penalized estimators of the nuisance functions.
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