Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

Mots-clés: Inhomogeneous Poisson process; change-point; Bayesian estimator; maximum likelihood estimator; likelihood ratio process

This work is devoted to the problem of change-point parameter estimation in the case of the presence of multiple changes in the intensity function of the Poisson process. It is supposed that the observations are independent inhomogeneous Poisson processes with the same intensity function and this intensity function has two jumps separated by a known quantity. The asymptotic behavior of the maximum-likelihood and Bayesian estimators are described. It is shown that these estimators are consistent, have different limit distributions, the moments converge and that the Bayesian estimators are asymptotically efficient. The  numerical simulations illustrate the obtained results.

We will talk brievely about a detection problem where several spatially distributed sensors observe Poisson signals emitted from a single radioactive source of unknown position. The measurements at each sensor are modeled by independent inhomogeneous Poisson processes. A method based on Bayesian change-point estimation is proposed to identify the location of the source’s coordinates. The asymptotic behavior of the Bayesian estimator is studied. In particular, the consistency and the asymptotic efficiency of the estimator are analyzed. The limit distribution and the convergence of the moments are also described. The similar statistical model could be used in GPS localization problems.

Références:
[1] Ibragimov, I. A., and Khasminskii, R. Z.,(1981) Statistical Estimation. Asymptotic Theory,
Springer, New York.
[2] Kutoyants, Yu. A.,(1998) Statistical Inference for Spatial Poisson Processes, Lecture Notes in
Statistics 134, Springer-Verlag, New York.