Electronic Journal of Statistics

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

Patricia Reynaud-Bouret and Vincent Rivoirard

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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 ℬαp,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 ℬαp,q with p2 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.

Article information

Electron. J. Statist. Volume 4 (2010), 172-238.

First available in Project Euclid: 11 February 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 62G20: Asymptotic properties

Adaptive estimation calibration model selection oracle inequalities Poisson process thresholding rule


Reynaud-Bouret, Patricia; Rivoirard, Vincent. Near optimal thresholding estimation of a Poisson intensity on the real line. Electron. J. Statist. 4 (2010), 172--238. doi:10.1214/08-EJS319. http://projecteuclid.org/euclid.ejs/1265899072.

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