Intensity estimation of non-homogeneous Poisson processes from shifted trajectories

Jérémie Bigot (1) , Sébastien Gadat (1) , Thierry Klein (1) , Clément Marteau (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

This paper considers the problem of adaptive estimation of a non-homogeneous intensity function from the observation of n independent Poisson processes having a common intensity that is randomly shifted for each observed trajectory. 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.

Mots clés: Poisson processes; Random shifts; Intensity estimation; Deconvolution; Meyer wavelets; Adaptive estimation; Besov space; Minimax rate

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Jérémie Bigot, Sébastien Gadat, Thierry Klein, Clément Marteau. Intensity estimation of non-homogeneous Poisson processes from shifted trajectories. 2011. ⟨hal-00593922⟩