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April 2018 Oracle inequalities for sparse additive quantile regression in reproducing kernel Hilbert space
Shaogao Lv, Huazhen Lin, Heng Lian, Jian Huang
Ann. Statist. 46(2): 781-813 (April 2018). DOI: 10.1214/17-AOS1567

Abstract

This paper considers the estimation of the sparse additive quantile regression (SAQR) in high-dimensional settings. Given the nonsmooth nature of the quantile loss function and the nonparametric complexities of the component function estimation, it is challenging to analyze the theoretical properties of ultrahigh-dimensional SAQR. We propose a regularized learning approach with a two-fold Lasso-type regularization in a reproducing kernel Hilbert space (RKHS) for SAQR. We establish nonasymptotic oracle inequalities for the excess risk of the proposed estimator without any coherent conditions. If additional assumptions including an extension of the restricted eigenvalue condition are satisfied, the proposed method enjoys sharp oracle rates without the light tail requirement. In particular, the proposed estimator achieves the minimax lower bounds established for sparse additive mean regression. As a by-product, we also establish the concentration inequality for estimating the population mean when the general Lipschitz loss is involved. The practical effectiveness of the new method is demonstrated by competitive numerical results.

Citation

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Shaogao Lv. Huazhen Lin. Heng Lian. Jian Huang. "Oracle inequalities for sparse additive quantile regression in reproducing kernel Hilbert space." Ann. Statist. 46 (2) 781 - 813, April 2018. https://doi.org/10.1214/17-AOS1567

Information

Received: 1 February 2016; Revised: 1 January 2017; Published: April 2018
First available in Project Euclid: 3 April 2018

zbMATH: 06870279
MathSciNet: MR3782384
Digital Object Identifier: 10.1214/17-AOS1567

Subjects:
Primary: 62G20
Secondary: 62G05

Keywords: Additive models , Quantile regression , regularization methods , ‎reproducing kernel Hilbert ‎space , Sparsity

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 2 • April 2018
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