文章基本信息

标题：Intensity estimation of non-homogeneous Poisson processes from shifted trajectories
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作者：Jérémie Bigot ; Sébastien Gadat ; Thierry Klein 等
期刊名称：Electronic Journal of Statistics
印刷版ISSN：1935-7524
出版年度：2013
卷号：7
页码：881-931
DOI：10.1214/13-EJS794
语种：English
出版社：Institute of Mathematical Statistics
摘要：In this paper, we consider the problem of estimating nonparametrically a mean pattern intensity $\lambda$ from the observation of $n$ independent and non-homogeneous Poisson processes $N^{1},\dots,N^{n}$ on the interval $[0,1]$. This problem arises when data (counts) are collected independently from $n$ individuals according to similar Poisson processes. We show that estimating this intensity is a deconvolution problem for which the density of the random shifts plays the role of the convolution operator. In an asymptotic setting where the number $n$ of observed trajectories tends to infinity, we derive upper and lower bounds for the minimax quadratic risk over Besov balls. Non-linear thresholding in a Meyer wavelet basis is used to derive an adaptive estimator of the intensity. The proposed estimator is shown to achieve a near-minimax rate of convergence. This rate depends both on the smoothness of the intensity function and the density of the random shifts, which makes a connection between the classical deconvolution problem in nonparametric statistics and the estimation of a mean intensity from the observations of independent Poisson processes.