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Intensity estimation of non-homogeneous Poisson processes from shifted trajectories. (English) Zbl 1337.62200
Summary: 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.

62M05 Markov processes: estimation; hidden Markov models
62G05 Nonparametric estimation
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
WaveD; WaveLab
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