Estimation of the invariant measure of a multidimensional diffusion from noisy observations
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914759982448640 |
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| author | Maillet, Raphaël Szymanski, Grégoire |
| author_facet | Maillet, Raphaël Szymanski, Grégoire |
| contents | We introduce a new approach for estimating the invariant density of a multidimensional diffusion when dealing with high-frequency observations blurred by independent noises. We consider the intermediate regime, where observations occur at discrete time instances $kΔ_n$ for $k=0,\dots,n$, under the conditions $Δ_n\to 0$ and $nΔ_n\to\infty$. Our methodology involves the construction of a kernel density estimator that uses a pre-averaging technique to proficiently remove noise from the data while preserving the analytical characteristics of the underlying signal and its asymptotic properties. The rate of convergence of our estimator depends on both the anisotropic regularity of the density and the intensity of the noise. We establish conditions on the intensity of the noise that ensure the recovery of convergence rates similar to those achievable without any noise. Furthermore, we prove a Bernstein concentration inequality for our estimator, from which we derive an adaptive procedure for the kernel bandwidth selection. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_12181 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Estimation of the invariant measure of a multidimensional diffusion from noisy observations Maillet, Raphaël Szymanski, Grégoire Statistics Theory Probability Methodology 62G05, 62G07, 62G20, 60J60 We introduce a new approach for estimating the invariant density of a multidimensional diffusion when dealing with high-frequency observations blurred by independent noises. We consider the intermediate regime, where observations occur at discrete time instances $kΔ_n$ for $k=0,\dots,n$, under the conditions $Δ_n\to 0$ and $nΔ_n\to\infty$. Our methodology involves the construction of a kernel density estimator that uses a pre-averaging technique to proficiently remove noise from the data while preserving the analytical characteristics of the underlying signal and its asymptotic properties. The rate of convergence of our estimator depends on both the anisotropic regularity of the density and the intensity of the noise. We establish conditions on the intensity of the noise that ensure the recovery of convergence rates similar to those achievable without any noise. Furthermore, we prove a Bernstein concentration inequality for our estimator, from which we derive an adaptive procedure for the kernel bandwidth selection. |
| title | Estimation of the invariant measure of a multidimensional diffusion from noisy observations |
| topic | Statistics Theory Probability Methodology 62G05, 62G07, 62G20, 60J60 |
| url | https://arxiv.org/abs/2404.12181 |