Near optimal thresholding estimation of a Poisson intensity on the real line. (English) Zbl 1329.62176

Summary: 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 \[ \sup_{f\in \mathcal B^{\alpha}_{p,q}\cap \mathbb L_{\infty}} \mathbb E\|\tilde f_{n, \gamma}-f\|{2\atop 2} \leq C\bigg(\frac{\log n}{n}\bigg)^{\frac{\alpha}{\alpha+\frac{1}{2}+\left(\frac{1}{2}-\frac{1}{p}\right)_{+}}} \] 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 \(p\leq 2\) 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 \(\gamma \) whose optimal value is hard to estimate from the data. In this paper, we study the problem of calibrating \(\gamma\) both theoretically and practically. Finally, some simulations are provided, proving the excellent practical behavior of our procedure with respect to the support issue.


62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
Full Text: DOI arXiv Euclid


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