Minimax estimation of a functional on a structured high-dimensional model

Abstract

We introduce a new method of estimation of parameters in semiparametric and nonparametric models. The method employs $U$-statistics that are based on higher-order influence functions of the parameter of interest, which extend ordinary linear influence functions, 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 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.