%0 Journal Article %1 bigot2013 %A Bigot, Jérémie %A Gadat, Sébastien %A Klein, Thierry %A Marteau, Clément %D 2013 %I The Institute of Mathematical Statistics and the Bernoulli Society %J Electron. J. Statist. %K Poisson_process Poisson_regression deconvolution inhomogeneous_Poisson_process statistics wavelets %P 881--931 %R 10.1214/13-EJS794 %T Intensity estimation of non-homogeneous Poisson processes from shifted trajectories %U http://dx.doi.org/10.1214/13-EJS794 %V 7 %X In this paper, we consider the problem of estimating nonpara- metrically a mean pattern intensity λ from the observation of n independent and non-homogeneous Poisson processes N 1 , . . . , 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 estimat- ing 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.

@article{bigot2013, abstract = {In this paper, we consider the problem of estimating nonpara- metrically a mean pattern intensity λ from the observation of n independent and non-homogeneous Poisson processes N 1 , . . . , 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 estimat- ing 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.}, added-at = {2015-02-18T22:03:24.000+0100}, author = {Bigot, Jérémie and Gadat, Sébastien and Klein, Thierry and Marteau, Clément}, biburl = {https://www.bibsonomy.org/bibtex/28497a74391066f6e88f5306e23eae715/peter.ralph}, doi = {10.1214/13-EJS794}, fjournal = {Electronic Journal of Statistics}, interhash = {7feb3d0f1c370b4513e7927ec0449f8e}, intrahash = {8497a74391066f6e88f5306e23eae715}, journal = {Electron. J. Statist.}, keywords = {Poisson_process Poisson_regression deconvolution inhomogeneous_Poisson_process statistics wavelets}, pages = {881--931}, publisher = {The Institute of Mathematical Statistics and the Bernoulli Society}, timestamp = {2015-02-18T22:03:24.000+0100}, title = {Intensity estimation of non-homogeneous Poisson processes from shifted trajectories}, url = {http://dx.doi.org/10.1214/13-EJS794}, volume = 7, year = 2013 }