Abstract
This paper develops a foundation of methodology and theory for nonparametric regression with Lie group-valued predictors contaminated by measurement errors. Our methodology and theory are based on harmonic analysis on Lie groups, which is largely unknown in statistics. We establish a novel deconvolution regression estimator, and study its rate of convergence and asymptotic distribution. We also provide asymptotic confidence intervals based on the asymptotic distribution of the estimator and on the empirical likelihood technique. Several theoretical properties are also studied for a deconvolution density estimator, which is necessary to construct our regression estimator. The case of unknown measurement error distribution is also covered. We present practical details on implementation as well as the results of simulation studies for several Lie groups. A real data example is also provided.
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 would like to thank an Associate Editor and four referees for helpful and constructive comments, which led to significant improvement of our paper.
Citation
Jeong Min Jeon. Byeong U. Park. Ingrid Van Keilegom. "Nonparametric regression on Lie groups with measurement errors." Ann. Statist. 50 (5) 2973 - 3008, October 2022. https://doi.org/10.1214/22-AOS2218
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