Electronic Journal of Statistics

Intensity estimation of non-homogeneous Poisson processes from shifted trajectories

Jérémie Bigot, Sébastien Gadat, Thierry Klein, and Clément Marteau

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Abstract

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.

Article information

Source
Electron. J. Statist., Volume 7 (2013), 881-931.

Dates
First available in Project Euclid: 3 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1364994252

Digital Object Identifier
doi:10.1214/13-EJS794

Mathematical Reviews number (MathSciNet)
MR3044503

Zentralblatt MATH identifier
1337.62200

Subjects
Primary: 62G08: Nonparametric regression
Secondary: 42C40: Wavelets and other special systems

Keywords
Poisson processes random shifts intensity estimation deconvolution Meyer wavelets adaptive estimation Besov space minimax rate

Citation

Bigot, Jérémie; Gadat, Sébastien; Klein, Thierry; Marteau, Clément. Intensity estimation of non-homogeneous Poisson processes from shifted trajectories. Electron. J. Statist. 7 (2013), 881--931. doi:10.1214/13-EJS794. https://projecteuclid.org/euclid.ejs/1364994252


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