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October 2017 Minimax estimation of a functional on a structured high-dimensional model
James M. Robins, Lingling Li, Rajarshi Mukherjee, Eric Tchetgen Tchetgen, Aad van der Vaart
Ann. Statist. 45(5): 1951-1987 (October 2017). DOI: 10.1214/16-AOS1515

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.

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James M. Robins. Lingling Li. Rajarshi Mukherjee. Eric Tchetgen Tchetgen. Aad van der Vaart. "Minimax estimation of a functional on a structured high-dimensional model." Ann. Statist. 45 (5) 1951 - 1987, October 2017. https://doi.org/10.1214/16-AOS1515

Information

Received: 1 April 2015; Revised: 1 August 2016; Published: October 2017
First available in Project Euclid: 31 October 2017

zbMATH: 06821115
MathSciNet: MR3718158
Digital Object Identifier: 10.1214/16-AOS1515

Subjects:
Primary: 62F25 , 62G05 , 62G20

Keywords: $U$-statistic , influence function , Nonlinear functional , nonparametric estimation , tangent space

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 5 • October 2017
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