Near optimal thresholding estimation of a Poisson intensity on the real line

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 possible logarithmic term. Then, minimax properties of $\tilde{f}_{n,\gamma}$ on Besov spaces $\mathcal{B}^{\alpha}_{p,q}$ are established. Under mild assumptions, we prove that $$\sup_{f\in \mathcal{B}^{\alpha}_{p,q}\cap \mathbb{L}_{\infty}}\mathbb{E}\|\tilde{f}_{n,\gamma}-f\|^{2}\leq C\left(\frac{ {\log \,}n}{n}\right)^{\frac{\alpha}{\alpha+\frac{1}{2}+\bigl(\frac{1}{2}-\frac{1}{p}\bigr)_{+}}}$$ and the lower bound of the minimax risk for $\mathcal{B}^{\alpha}_{p,q}\cap \mathbb{L}_{\infty}$ 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 $\mathcal{B}^{\alpha}_{p,q}$ with $p≤2$ 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, $\tilde{f}_{n,\gamma}$ is adaptive minimax up to a logarithmic term. Our procedure is based on data-driven thresholds. As usual, they depend on a tuning parameter $γ$ whose optimal value is hard to estimate from the data. In this paper, we study the problem of calibrating $γ$ both theoretically and practically. Finally, some simulations are provided, proving the excellent practical behavior of our procedure with respect to the support issue.