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October 2017 Nonasymptotic analysis of semiparametric regression models with high-dimensional parametric coefficients
Ying Zhu
Ann. Statist. 45(5): 2274-2298 (October 2017). DOI: 10.1214/16-AOS1528

Abstract

We consider a two-step projection based Lasso procedure for estimating a partially linear regression model where the number of coefficients in the linear component can exceed the sample size and these coefficients belong to the $l_{q}$-“balls” for $q\in[0,1]$. Our theoretical results regarding the properties of the estimators are nonasymptotic. In particular, we establish a new nonasymptotic “oracle” result: Although the error of the nonparametric projection per se (with respect to the prediction norm) has the scaling $t_{n}$ in the first step, it only contributes a scaling $t_{n}^{2}$ in the $l_{2}$-error of the second-step estimator for the linear coefficients. This new “oracle” result holds for a large family of nonparametric least squares procedures and regularized nonparametric least squares procedures for the first-step estimation and the driver behind it lies in the projection strategy. We specialize our analysis to the estimation of a semiparametric sample selection model and provide a simple method with theoretical guarantees for choosing the regularization parameter in practice.

Citation

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Ying Zhu. "Nonasymptotic analysis of semiparametric regression models with high-dimensional parametric coefficients." Ann. Statist. 45 (5) 2274 - 2298, October 2017. https://doi.org/10.1214/16-AOS1528

Information

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

zbMATH: 06821126
MathSciNet: MR3718169
Digital Object Identifier: 10.1214/16-AOS1528

Subjects:
Primary: 62J02
Secondary: 62G08, 62J12, 62N01, 62N02

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 5 • October 2017
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