From Isotonic to Lipschitz Regression: A New Interpolative Perspective on Shape-restricted Estimation
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866911702658842624 |
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| author | Takatsu, Kenta Zhang, Tianyu Kuchibhotla, Arun Kumar |
| author_facet | Takatsu, Kenta Zhang, Tianyu Kuchibhotla, Arun Kumar |
| contents | This manuscript bridges nonparametric smoothness-based and shape-restricted estimation, which may appear as two disjoint paradigms in the field. The proposed approach is motivated by a conceptually simple observation: every Lipschitz function is a sum of a monotonic and a linear function. This principle is further generalized to the higher-order monotonicity and multivariate settings. A family of estimators is proposed based on a sample-splitting procedure, inheriting desirable methodological, theoretical, and computational properties of shape-restricted estimators. The theoretical analysis provides convergence guarantees of the estimator under heteroscedastic and heavy-tailed errors, as well as adaptivity to the unknown ``complexity" of the true regression function. The generality of the proposed decomposition framework is demonstrated through new approximation results and numerical studies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_05732 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | From Isotonic to Lipschitz Regression: A New Interpolative Perspective on Shape-restricted Estimation Takatsu, Kenta Zhang, Tianyu Kuchibhotla, Arun Kumar Methodology Statistics Theory This manuscript bridges nonparametric smoothness-based and shape-restricted estimation, which may appear as two disjoint paradigms in the field. The proposed approach is motivated by a conceptually simple observation: every Lipschitz function is a sum of a monotonic and a linear function. This principle is further generalized to the higher-order monotonicity and multivariate settings. A family of estimators is proposed based on a sample-splitting procedure, inheriting desirable methodological, theoretical, and computational properties of shape-restricted estimators. The theoretical analysis provides convergence guarantees of the estimator under heteroscedastic and heavy-tailed errors, as well as adaptivity to the unknown ``complexity" of the true regression function. The generality of the proposed decomposition framework is demonstrated through new approximation results and numerical studies. |
| title | From Isotonic to Lipschitz Regression: A New Interpolative Perspective on Shape-restricted Estimation |
| topic | Methodology Statistics Theory |
| url | https://arxiv.org/abs/2307.05732 |