Intensity estimation of non-homogeneous Poisson processes from shifted trajectories

TL;DR

This paper addresses the problem of estimating a non-homogeneous Poisson process intensity from shifted trajectories, framing it as a deconvolution problem and proposing an adaptive wavelet-based estimator with near-minimax convergence.

Contribution

It introduces a novel approach linking intensity estimation to deconvolution, providing theoretical bounds and an adaptive estimator for non-homogeneous Poisson processes.

Findings

Derived upper and lower bounds for minimax risk

Proposed an adaptive wavelet-based estimator

Achieved near-minimax convergence rate

Abstract

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.