In this article we study a class of parameters with the so-called `mixed bias property'. For parameters with this property, the bias of the semiparametric efficient one step estimator is equal to the mean of the product of the estimation errors of two nuisance functions. In non-parametric models, parameters with the mixed bias property admit so-called rate doubly robust estimators, i.e. estimators that are 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 the estimator of one of the nuisance functions with faster rates for the estimator of the other nuisance. We show that the class of parameters with the mixed bias property strictly includes two recently studied classes of parameters which, in turn, include many parameters of interest in causal inference. We characterize the form of parameters with the mixed bias property and of their influence functions. Furthermore, we derive two functional moment equations, each being solved at one of the two nuisance functions, as well as, two functional loss functions, each being minimized at one of the two nuisance functions. These loss functions can be used to derive loss based penalized estimators of the nuisance functions.

中文翻译：

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具有混合偏置特性的参数表征

在本文中，我们研究了一类具有所谓“混合偏置特性”的参数。对于具有此属性的参数，半参数有效一步估计量的偏差等于两个干扰函数的估计误差乘积的均值。在非参数模型中，具有混合偏差属性的参数允许所谓的速率双稳健估计量，即当成功地以足够快的速率估计两个干扰函数时一致且渐近正态的估计量，并有可能折衷较慢的速率一个干扰函数的估计器的收敛速度更快，另一个干扰函数的估计器的收敛速度更快。我们表明具有混合偏置属性的参数类严格包括两个最近研究的参数类，反过来，包括许多因果推断中感兴趣的参数。我们用混合偏置属性及其影响函数来表征参数的形式。此外，我们推导出两个函数矩方程，每个方程在两个干扰函数之一上求解，以及两个函数损失函数，每个在两个干扰函数之一处最小化。这些损失函数可用于导出基于损失的有害函数的惩罚估计量。