Asymptotic Equivalence for Nonparametric Generalized Linear Models
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915071615041536 |
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| author | Grama, Ion Nussbaum, Michael |
| author_facet | Grama, Ion Nussbaum, Michael |
| contents | We establish that a non-Gaussian nonparametric regression model is asymptotically equivalent to a regression model with Gaussian noise. The approximation is in the sense of Le Cam's deficiency distance $Δ$; the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. Our result concerns a sequence of independent but not identically distributed observations with each distribution in the same real-indexed exponential family. The canonical parameter is a value $f(t_i)$ of a regression function $f$ at a grid point $t_i$ (nonparametric GLM). When $f$ is in a Hölder ball with exponent $β>\frac 12 ,$ we establish global asymptotic equivalence to observations of a signal $Γ(f(t))$ in Gaussian white noise, where $Γ$ is related to a variance stabilizing transformation in the exponential family. The result is a regression analog of the recently established Gaussian approximation for the i.i.d. model. The proof is based on a functional version of the Hungarian construction for the partial sum process. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15057 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic Equivalence for Nonparametric Generalized Linear Models Grama, Ion Nussbaum, Michael Statistics Theory We establish that a non-Gaussian nonparametric regression model is asymptotically equivalent to a regression model with Gaussian noise. The approximation is in the sense of Le Cam's deficiency distance $Δ$; the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. Our result concerns a sequence of independent but not identically distributed observations with each distribution in the same real-indexed exponential family. The canonical parameter is a value $f(t_i)$ of a regression function $f$ at a grid point $t_i$ (nonparametric GLM). When $f$ is in a Hölder ball with exponent $β>\frac 12 ,$ we establish global asymptotic equivalence to observations of a signal $Γ(f(t))$ in Gaussian white noise, where $Γ$ is related to a variance stabilizing transformation in the exponential family. The result is a regression analog of the recently established Gaussian approximation for the i.i.d. model. The proof is based on a functional version of the Hungarian construction for the partial sum process. |
| title | Asymptotic Equivalence for Nonparametric Generalized Linear Models |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2412.15057 |