We introduce a new method of estimation of parameters in semiparametric and nonparametric models. The method is based on estimating equations that are $U$-statistics in the observations. The $U$-statistics 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 leads to a bias-variance trade-off, and results in estimators that converge at a slower than $\sqrt n$-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 $\sqrt n$-rate, but we also consider efficient $\sqrt n$-estimation using novel nonlinear estimators. The general approach is applied in detail to the example of estimating a mean response when the response is not always observed.

翻译：我们提出了一种在半参数和非参数模型中估计参数的新方法。该方法基于观测数据中的$U$-统计量构成的估计方程。这些$U$-统计量基于高阶影响函数，这些函数推广了目标参数的普通线性影响函数，并代表了该参数的高阶导数。对于表示无法完全实现的参数，该方法会导致偏差-方差权衡，并产生以慢于$\sqrt n$速率收敛的估计量。在许多示例中，所得速率可以证明是最优的。我们特别关注在具有高维或低正则性 nuisance 参数的模型中估计参数的情况，其中目标参数无法以$\sqrt n$速率进行估计，但我们也考虑了使用新颖非线性估计量的高效$\sqrt n$估计。这种通用方法被详细应用于当响应变量并非始终被观测到时估计平均响应的示例。