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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.

##### MSC:
 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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