Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2410.11777 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917191013629952 |
|---|---|
| author | Divol, Vincent Guérin, Hélène Nguyen, Dinh-Toan Tran, Viet Chi |
| author_facet | Divol, Vincent Guérin, Hélène Nguyen, Dinh-Toan Tran, Viet Chi |
| contents | From the observation of a diffusion path $(X_t)_{t\in [0,T]}$ on a compact connected $d$-dimensional manifold $\mathcal{M}$ without boundary, we consider the problem of estimating the stationary measure $μ$ of the process. Wang and Zhu (2023) showed that for the Wasserstein metric $\mathcal{W}_2$ and for $d\geq 5$, the convergence rate of $T^{-1/(d-2)}$ is attained by the occupation measure of the path $(X_t)_{t\in [0,T]}$ when $(X_t)_{t\in [0,T]}$ is a Langevin diffusion. We extend their result in several directions. First, we show that the rate of convergence holds for a large class of diffusion paths, whose generators are uniformly elliptic. Second, the regularity of the density $p$ of the stationary measure $μ$ with respect to the volume measure of $\mathcal{M}$ can be leveraged to obtain faster estimators: when $p$ belongs to a Sobolev space of order $\ell\geq 2$, smoothing the occupation measure by convolution with a kernel yields an estimator whose rate of convergence is of order $T^{-(\ell+1)/(2\ell+d-2)}$. We further show that this rate is the minimax rate of estimation for this problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11777 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Measure estimation on a manifold explored by a diffusion process Divol, Vincent Guérin, Hélène Nguyen, Dinh-Toan Tran, Viet Chi Statistics Theory Probability 60F17, 05C81, 62G07 From the observation of a diffusion path $(X_t)_{t\in [0,T]}$ on a compact connected $d$-dimensional manifold $\mathcal{M}$ without boundary, we consider the problem of estimating the stationary measure $μ$ of the process. Wang and Zhu (2023) showed that for the Wasserstein metric $\mathcal{W}_2$ and for $d\geq 5$, the convergence rate of $T^{-1/(d-2)}$ is attained by the occupation measure of the path $(X_t)_{t\in [0,T]}$ when $(X_t)_{t\in [0,T]}$ is a Langevin diffusion. We extend their result in several directions. First, we show that the rate of convergence holds for a large class of diffusion paths, whose generators are uniformly elliptic. Second, the regularity of the density $p$ of the stationary measure $μ$ with respect to the volume measure of $\mathcal{M}$ can be leveraged to obtain faster estimators: when $p$ belongs to a Sobolev space of order $\ell\geq 2$, smoothing the occupation measure by convolution with a kernel yields an estimator whose rate of convergence is of order $T^{-(\ell+1)/(2\ell+d-2)}$. We further show that this rate is the minimax rate of estimation for this problem. |
| title | Measure estimation on a manifold explored by a diffusion process |
| topic | Statistics Theory Probability 60F17, 05C81, 62G07 |
| url | https://arxiv.org/abs/2410.11777 |