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| Auteurs principaux: | , |
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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2501.04257 |
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| _version_ | 1866909451128143872 |
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| author | Sanchez, Claudia Fonte Hoffmann, Marc |
| author_facet | Sanchez, Claudia Fonte Hoffmann, Marc |
| contents | We consider an interacting system of particles with value in $\mathbb{R}^d \times \mathbb{R}^d$, governed by transport and diffusion on the first component, on that may serve as a representative model for kinetic models with a degenerate component. In a first part, we control the fluctuations of the empirical measure of the system around the solution of the corresponding Vlasov-Fokker-Planck equation by proving a Bernstein concentration inequality, extending a previous result of arXiv:2011.03762 in several directions. In a second part, we study the nonparametric statistical estimation of the classical solution of Vlasov-Fokker-Planck equation from the observation of the empirical measure and prove an oracle inequality using the Goldenshluger-Lepski methodology and we obtain minimax optimality. We then specialise on the FitzHugh-Nagumo model for populations of neurons. We consider a version of the model proposed in Mischler et al. arXiv:1503.00492 an optimally estimate the $6$ parameters of the model by moment estimators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04257 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Statistical estimation of a mean-field FitzHugh-Nagumo model Sanchez, Claudia Fonte Hoffmann, Marc Statistics Theory We consider an interacting system of particles with value in $\mathbb{R}^d \times \mathbb{R}^d$, governed by transport and diffusion on the first component, on that may serve as a representative model for kinetic models with a degenerate component. In a first part, we control the fluctuations of the empirical measure of the system around the solution of the corresponding Vlasov-Fokker-Planck equation by proving a Bernstein concentration inequality, extending a previous result of arXiv:2011.03762 in several directions. In a second part, we study the nonparametric statistical estimation of the classical solution of Vlasov-Fokker-Planck equation from the observation of the empirical measure and prove an oracle inequality using the Goldenshluger-Lepski methodology and we obtain minimax optimality. We then specialise on the FitzHugh-Nagumo model for populations of neurons. We consider a version of the model proposed in Mischler et al. arXiv:1503.00492 an optimally estimate the $6$ parameters of the model by moment estimators. |
| title | Statistical estimation of a mean-field FitzHugh-Nagumo model |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2501.04257 |