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

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

Patricia Reynaud-Bouret and Vincent Rivoirard

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

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 ℬ^α_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 ℬ^α_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.

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Primary Subjects: 62G05

Secondary Subjects: 62G20

Keywords: Adaptive estimation; calibration; model selection; oracle inequalities; Poisson process; thresholding rule

Full-text: Open access

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Permanent link to this document: http://projecteuclid.org/euclid.ejs/1265899072
Digital Object Identifier: doi:10.1214/08-EJS319
Mathematical Reviews number (MathSciNet): MR2645482

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