标题： 具有无限支撑的泊松强度的自适应阈值估计 显示英文标题

标题： Adaptive thresholding estimation of a Poisson intensity with infinite support

摘要： 本文的目的是利用阈值规则估计泊松过程的强度$N$。在本文中，假设强度被定义为$N$的均值测度关于$ndx$的导数，其中$n$是固定参数，并且该强度被认为是非紧支集的。基于随机阈值的估计量$\tilde{f}_{n,\gamma}$被证明在对数项内达到与理想估计量相同的性能。通过oracle不等式可以推导出$\tilde{f}_{n,\gamma}$的maxiset。然后，建立了$\tilde{f}_{n,\gamma}$的minimax性质。 我们首先证明，当 $p\leq 2$ 是 $(\ln(n)/n)^{\al/(1+2\al)}$ 时，该估计量在 Besov 空间 ${\cal B}^\al_{p,q}$ 上的收敛速度。 这一结果有两个推论。 首先，它表明，当考虑非紧支函数时，Besov 空间 ${\cal B}^\al_{p,q}$ 的 minimax 收敛速度（其中 $p\leq 2$）与紧支函数的 minimax 收敛速度相同，相差一个对数项。 这个结果是新的。 此外， $\tilde{f}_{n,\gamma}$ 在对数项范围内是自适应 minimax 的。 当$p>2$时，情况发生了剧烈变化，且$\tilde{f}_{n,\gamma}$在 Besov 空间${\cal B}^\al_{p,q}$上的表现比$(\ln(n)/n)^{\al/(1+2\al)}$要差。最后，随机阈值依赖于一个需要在实际应用中适当选择的参数$\gamma$。一些理论结果提供了$\gamma$的上下界，以获得令人满意的oracle不等式。模拟实验加强了这些结果。 显示英文摘要

摘要： 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.