This paper considers adaptive estimation of quadratic functionals in the nonparametric instrumental variables (NPIV) models. Minimax estimation of a quadratic functional of a NPIV is an important problem in optimal estimation of a nonlinear functional of an ill-posed inverse regression with an unknown operator using one random sample. We first show that a leave-one-out, sieve NPIV estimator of the quadratic functional proposed by \cite{BC2020} attains a convergence rate that coincides with the lower bound previously derived by \cite{ChenChristensen2017}. The minimax rate is achieved by the optimal choice of a key tuning parameter (sieve dimension) that depends on unknown NPIV model features. We next propose a data driven choice of the tuning parameter based on Lepski's method. The adaptive estimator attains the minimax optimal rate in the severely ill-posed case and in the regular, mildly ill-posed case, but up to a multiplicative $\sqrt{\log n}$ in the irregular, mildly ill-posed case.
翻译:本文考虑了非参数工具变量( NPIV) 模型中的二次功能的适应性估计。 NPIV 模型的二次函数的最小值估计是最佳估计一个未知操作者使用随机样本与未知操作者进行不测的反向回归的非线性功能的一个重要问题。 我们首先显示,\ cite{BC20} 提议的二次函数的静脉 NPIV 估计值在离线后达到与先前由\cite{Chen Christensen2017} 得出的较低约束值相符的趋同率。 迷你速是通过最佳选择关键调控参数( 缩略度) 实现的, 取决于未知 NPIV 模型的特性。 我们接下来提议根据 Lepski 方法, 由数据驱动选择调控参数。 适应性估计器在严重错误的情况下和常规、 轻度错误的情况下, 达到微轴值最佳速率, 但在正常、 轻度情况下, 最高为多复制 $sqrgnpos} 。