Adaptive thresholding estimation of a Poisson intensity with infinite support
| dc.creator | Reynaud-Bouret, Patricia | |
| dc.creator | Rivoirard, Vincent | |
| dc.date | 2008-01-21 | |
| dc.date.accessioned | 2026-07-07T08:55:37Z | |
| dc.date.available | 2026-07-07T08:55:37Z | |
| dc.description | The purpose of this paper is to estimate the intensity of a Poisson process $N$ by using thresholding rules. In this paper, the intensity, defined as the derivative of the mean measure of $N$ with respect to $ndx$ where $n$ is a fixed parameter, is assumed to be non-compactly supported. The estimator $\tilde{f}_{n,γ}$ based on random thresholds is proved to achieve the same performance as the oracle estimator up to a logarithmic term. Oracle inequalities allow to derive the maxiset of $\tilde{f}_{n,γ}$. Then, minimax properties of $\tilde{f}_{n,γ}$ are established. We first prove that the rate of this estimator on Besov spaces ${\cal B}^\al_{p,q}$ when $p\leq 2$ is $(\ln(n)/n)^{\al/(1+2\al)}$. This result has two consequences. First, it establishes that the minimax rate of Besov spaces ${\cal B}^\al_{p,q}$ with $p\leq 2$ when non compactly supported functions are considered is the same as for compactly supported functions up to a logarithmic term. This result is new. Furthermore, $\tilde{f}_{n,γ}$ is adaptive minimax up to a logarithmic term. When $p>2$, the situation changes dramatically and the rate of $\tilde{f}_{n,γ}$ on Besov spaces ${\cal B}^\al_{p,q}$ is worse than $(\ln(n)/n)^{\al/(1+2\al)}$. Finally, the random threshold depends on a parameter $γ$ that has to be suitably chosen in practice. Some theoretical results provide upper and lower bounds of $γ$ to obtain satisfying oracle inequalities. Simulations reinforce these results. | |
| dc.identifier | https://arxiv.org/abs/0801.3157 | |
| dc.identifier | http://arxiv.org/abs/0801.3157 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/146327 | |
| dc.subject | Statistics Theory | |
| dc.title | Adaptive thresholding estimation of a Poisson intensity with infinite support | |
| dc.type | text |