Varying-Coefficient Fr\'echet Regression

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.