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October 2021 Additive regression for non-Euclidean responses and predictors
Jeong Min Jeon, Byeong U. Park, Ingrid Van Keilegom
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Ann. Statist. 49(5): 2611-2641 (October 2021). DOI: 10.1214/21-AOS2048

Abstract

Additive regression is studied in a very general setting where both the response and predictors are allowed to be non-Euclidean. The response takes values in a general separable Hilbert space, whereas the predictors take values in general semimetric spaces, which covers a very wide range of nonstandard response variables and predictors. A general framework of estimating additive models is presented for semimetric space-valued predictors. In particular, full details of implementation and the corresponding theory are given for predictors taking values in Hilbert spaces and/or Riemannian manifolds. The existence of the estimators, convergence of a backfitting algorithm, rates of convergence and asymptotic distributions of the estimators are discussed. The finite sample performance of the estimators is investigated by means of two simulation studies. Finally, three data sets covering several types of non-Euclidean data are analyzed to illustrate the usefulness of the proposed general approach.

Funding Statement

The first author acknowledges financial support from the European Research Council (2016–2021, Horizon 2020/ERC grant agreement no. 694409) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2020R1A6A3A03037314). Research of the second author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2019R1A2C3007355). The third author acknowledges financial support from the European Research Council (2016–2021, Horizon 2020/ERC grant agreement no. 694409).

Acknowledgments

The authors thank an Associate Editor and two referees for giving thoughtful and constructive comments on the earlier versions of the paper.

The authors would like to thank Qiang Wu and Guillermo Henry for answering to some questions on their papers.

Citation

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Jeong Min Jeon. Byeong U. Park. Ingrid Van Keilegom. "Additive regression for non-Euclidean responses and predictors." Ann. Statist. 49 (5) 2611 - 2641, October 2021. https://doi.org/10.1214/21-AOS2048

Information

Received: 1 March 2020; Revised: 1 January 2021; Published: October 2021
First available in Project Euclid: 12 November 2021

Digital Object Identifier: 10.1214/21-AOS2048

Subjects:
Primary: 62G08
Secondary: 62G20

Keywords: Additive models , Compositional data , density-valued data , directional data , functional data , ‎Hilbert spaces , non-Euclidean data , Riemannian manifolds , shape data , smooth backfitting

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 5 • October 2021
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