Adaptive thresholding estimation of a Poisson intensity with infinite support

Abstract

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 $p\le 2$ 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 $p\le 2$ 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.