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.
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).
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.
"Additive regression for non-Euclidean responses and predictors." Ann. Statist. 49 (5) 2611 - 2641, October 2021. https://doi.org/10.1214/21-AOS2048