Adaptive Estimation of Quadratic Functionals in Nonparametric Instrumental Variable Models

• URL: http://arxiv.org/abs/2101.12282v1
• Date: Thu, 28 Jan 2021 21:14:02 GMT
• Title: Adaptive Estimation of Quadratic Functionals in Nonparametric Instrumental Variable Models
• Authors: Christoph Breunig, Xiaohong Chen
• Abstract summary: This paper considers adaptive estimation of quadratic functionals in the nonparametric instrumental variables (NPIV) models. We first show that a leave-one-out, sieve NPIV estimator attains a convergence rate that coincides with the lower bound. 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 $sqrtlog n$ in the irregular, mildly ill-posed case.
• Score: 1.6539154611511273
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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