ADAPTIVE ESTIMATION OF FUNCTIONALS IN NONPARAMETRIC INSTRUMENTAL REGRESSION
研究了在存在工具变量的非参数回归中，如何自适应地估计线性泛函的值，提出了一种基于降维和阈值化的插件估计器，并给出了数据驱动的参数选择方法，使估计达到最优收敛速度。
We consider the problem of estimating the value ℓ ( ϕ ) of a linear functional, where the structural function ϕ models a nonparametric relationship in presence of instrumental variables. We propose a plug-in estimator which is based on a dimension reduction technique and additional thresholding. It is shown that this estimator is consistent and can attain the minimax optimal rate of convergence under additional regularity conditions. This, however, requires an optimal choice of the dimension parameter m depending on certain characteristics of the structural function ϕ and the joint distribution of the regressor and the instrument, which are unknown in practice. We propose a fully data driven choice of m which combines model selection and Lepski’s method. We show that the adaptive estimator attains the optimal rate of convergence up to a logarithmic factor. The theory in this paper is illustrated by considering classical smoothness assumptions and we discuss examples such as pointwise estimation or estimation of averages of the structural function ϕ .