Mathematics – Statistics Theory
Scientific paper
2008-01-21
Mathematics
Statistics Theory
Scientific paper
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 $\tilde{f}_{n,\gamma}$ 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 $\tilde{f}_{n,\gamma}$. Then, minimax properties of $\tilde{f}_{n,\gamma}$ 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, $\tilde{f}_{n,\gamma}$ is adaptive minimax up to a logarithmic term. When $p>2$, the situation changes dramatically and the rate of $\tilde{f}_{n,\gamma}$ 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 $\gamma$ that has to be suitably chosen in practice. Some theoretical results provide upper and lower bounds of $\gamma$ to obtain satisfying oracle inequalities. Simulations reinforce these results.
Reynaud-Bouret Patricia
Rivoirard Vincent
No associations
LandOfFree
Adaptive thresholding estimation of a Poisson intensity with infinite support does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Adaptive thresholding estimation of a Poisson intensity with infinite support, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Adaptive thresholding estimation of a Poisson intensity with infinite support will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-594109