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May 2019 Minimax optimal estimation in partially linear additive models under high dimension
Zhuqing Yu, Michael Levine, Guang Cheng
Bernoulli 25(2): 1289-1325 (May 2019). DOI: 10.3150/18-BEJ1021


In this paper, we derive minimax rates for estimating both parametric and nonparametric components in partially linear additive models with high dimensional sparse vectors and smooth functional components. The minimax lower bound for Euclidean components is the typical sparse estimation rate that is independent of nonparametric smoothness indices. However, the minimax lower bound for each component function exhibits an interplay between the dimensionality and sparsity of the parametric component and the smoothness of the relevant nonparametric component. Indeed, the minimax risk for smooth nonparametric estimation can be slowed down to the sparse estimation rate whenever the smoothness of the nonparametric component or dimensionality of the parametric component is sufficiently large. In the above setting, we demonstrate that penalized least square estimators can nearly achieve minimax lower bounds.


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Zhuqing Yu. Michael Levine. Guang Cheng. "Minimax optimal estimation in partially linear additive models under high dimension." Bernoulli 25 (2) 1289 - 1325, May 2019.


Received: 1 June 2017; Revised: 1 January 2018; Published: May 2019
First available in Project Euclid: 6 March 2019

zbMATH: 07049407
MathSciNet: MR3920373
Digital Object Identifier: 10.3150/18-BEJ1021

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability


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Vol.25 • No. 2 • May 2019
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