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October 2019 Doubly penalized estimation in additive regression with high-dimensional data
Zhiqiang Tan, Cun-Hui Zhang
Ann. Statist. 47(5): 2567-2600 (October 2019). DOI: 10.1214/18-AOS1757


Additive regression provides an extension of linear regression by modeling the signal of a response as a sum of functions of covariates of relatively low complexity. We study penalized estimation in high-dimensional nonparametric additive regression where functional semi-norms are used to induce smoothness of component functions and the empirical $L_{2}$ norm is used to induce sparsity. The functional semi-norms can be of Sobolev or bounded variation types and are allowed to be different amongst individual component functions. We establish oracle inequalities for the predictive performance of such methods under three simple technical conditions: a sub-Gaussian condition on the noise, a compatibility condition on the design and the functional classes under consideration and an entropy condition on the functional classes. For random designs, the sample compatibility condition can be replaced by its population version under an additional condition to ensure suitable convergence of empirical norms. In homogeneous settings where the complexities of the component functions are of the same order, our results provide a spectrum of minimax convergence rates, from the so-called slow rate without requiring the compatibility condition to the fast rate under the hard sparsity or certain $L_{q}$ sparsity to allow many small components in the true regression function. These results significantly broaden and sharpen existing ones in the literature.


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Zhiqiang Tan. Cun-Hui Zhang. "Doubly penalized estimation in additive regression with high-dimensional data." Ann. Statist. 47 (5) 2567 - 2600, October 2019.


Received: 1 April 2017; Revised: 1 July 2018; Published: October 2019
First available in Project Euclid: 3 August 2019

zbMATH: 07114922
MathSciNet: MR3988766
Digital Object Identifier: 10.1214/18-AOS1757

Primary: 62E20, 62F25, 62F35
Secondary: 62J05, 62J12

Rights: Copyright © 2019 Institute of Mathematical Statistics


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Vol.47 • No. 5 • October 2019
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