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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.10482 |
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| _version_ | 1866915957530689536 |
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| author | He, Shuaida Chen, Yangzhou Chen, Xin |
| author_facet | He, Shuaida Chen, Yangzhou Chen, Xin |
| contents | Modern regression analysis often involves responses and predictors taking values in the same or distinct metric spaces. To rank non-Euclidean heterogeneous predictors in regression by explanatory strength, analogous to the classical $R^2$, we introduce the Fréchet correlation coefficient (FCC), defined as the relative reduction in the Fréchet variance of the response after conditioning on a specific predictor. FCC is directional, model-free, and interpretable on a unit-scale, attaining one under almost sure functional dependence and zero when the Fréchet mean is invariant to conditioning. We propose a novel partition-based estimator that avoids explicit nonparametric estimation of the conditional Fréchet mean function, thereby improving both computational efficiency and flexibility. A tailored wild bootstrap algorithm is further developed for testing the Fréchet conditional mean dependence. We establish asymptotic theory and evaluate power through extensive simulations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10482 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Fréchet correlation coefficient for heterogeneous random objects He, Shuaida Chen, Yangzhou Chen, Xin Methodology Modern regression analysis often involves responses and predictors taking values in the same or distinct metric spaces. To rank non-Euclidean heterogeneous predictors in regression by explanatory strength, analogous to the classical $R^2$, we introduce the Fréchet correlation coefficient (FCC), defined as the relative reduction in the Fréchet variance of the response after conditioning on a specific predictor. FCC is directional, model-free, and interpretable on a unit-scale, attaining one under almost sure functional dependence and zero when the Fréchet mean is invariant to conditioning. We propose a novel partition-based estimator that avoids explicit nonparametric estimation of the conditional Fréchet mean function, thereby improving both computational efficiency and flexibility. A tailored wild bootstrap algorithm is further developed for testing the Fréchet conditional mean dependence. We establish asymptotic theory and evaluate power through extensive simulations. |
| title | The Fréchet correlation coefficient for heterogeneous random objects |
| topic | Methodology |
| url | https://arxiv.org/abs/2604.10482 |