Abstract

As a growing number of problems involve variables that are random

objects, the development of models for such data has become increasingly important.

This paper introduces a novel varying-coefficient Fr´echet regression

model that extends the classical varying-coefficient framework to accommodate

random objects as responses. The proposed model provides a unified methodology for analyzing both Euclidean and non-Euclidean response variables. We

develop a comprehensive estimation procedure that accommodates diverse predictor settings. Specifically, the model allows the effect-modifier variable U to

be either Euclidean or non-Euclidean, while the predictors X are assumed to

be Euclidean. Tailored estimation methods are provided for each scenario. To

examine the asymptotic properties of the estimators, we introduce a smoothed

version of the model and establish convergence rates through separate theoretical

analyses of the bias and stochastic terms. The effectiveness and practical utility of the proposed methodology are demonstrated through extensive simulation

studies and a real-data application.

Key words and phrases: Fr´echet regression, metric space, random objects, varying-coefficient model

Information

Preprint No.SS-2025-0349
Manuscript IDSS-2025-0349
Complete AuthorsYanzhao Wang, Jianqiang Zhang, Wangli Xu
Corresponding AuthorsWangli Xu
Emailswlxu@ruc.edu.cn

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Acknowledgments

This work was supported by the National Key R&D Program of China

(Grant No. 2023YFA1008702) and the National Natural Science Foundation of China (Grant No. 12571300).

Supplementary Materials

This supplementary material contains additional implementation and simulation details, explanations and verifications of assumptions, and proofs of

the main results.

Supplementary materials are available for download.