Near optimal thresholding estimation of a Poisson intensity on the real line
| dc.creator | Reynaud-Bouret, Patricia | |
| dc.creator | Rivoirard, Vincent | |
| dc.date | 2008-10-29 | |
| dc.date.accessioned | 2026-07-07T10:13:51Z | |
| dc.date.available | 2026-07-07T10:13:51Z | |
| 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 possible logarithmic term. Then, minimax properties of $\tilde{f}_{n,γ}$ on Besov spaces ${\cal B}^{\ensuremath α}_{p,q}$ are established. Under mild assumptions, we prove that $$\sup_{f\in B^{\ensuremath α}_{p,q}\cap \ensuremath \mathbb {L}_{\infty}} \ensuremath \mathbb {E}(\ensuremath | | \tilde{f}_{n,γ}-f| |_2^2)\leq C(\frac{\log n}{n})^{\frac{\ensuremath α}{\ensuremath α+{1/2}+({1/2}-\frac{1}{p})_+}}$$ and the lower bound of the minimax risk for ${\cal B}^{\ensuremath α}_{p,q}\cap \ensuremath \mathbb {L}_{\infty}$ coincides with the previous upper bound up to the logarithmic term. This new result has two consequences. First, it establishes that the minimax rate of Besov spaces ${\cal B}^{\ensuremath α}_{p,q}$ with $p\leq 2$ when non compactly supported functions are considered is the same as for compactly supported functions up to a logarithmic term. When $p>2$, the rate exponent, which depends on $p$, deteriorates when $p$ increases, which means that the support plays a harmful role in this case. Furthermore, $\tilde{f}_{n,γ}$ is adaptive minimax up to a logarithmic term. | |
| dc.description | Submitted to the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org) | |
| dc.identifier | https://arxiv.org/abs/0810.5204 | |
| dc.identifier | http://arxiv.org/abs/0810.5204 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/172679 | |
| dc.subject | Statistics Theory | |
| dc.title | Near optimal thresholding estimation of a Poisson intensity on the real line | |
| dc.type | text |