Poisson Source Localization on the Plane. Smooth Case

TL;DR

This paper studies the problem of localizing a Poisson source on a plane using multiple detectors, analyzing estimators' properties and proposing efficient estimation methods.

Contribution

It introduces a comprehensive analysis of maximum likelihood and Bayesian estimators for Poisson source localization, and proposes efficient one-step MLE-processes.

Findings

MLE and Bayesian estimators are consistent and asymptotically normal.

Estimators are asymptotically efficient under regularity conditions.

Proposes simple consistent estimators and constructs efficient one-step MLE-processes.

Abstract

We consider the problem of localization of Poisson source by the observations of inhomogeneous Poisson processes. We suppose that there are $k$ detectors on the plane and each detector provides the observations of Poisson processes whose intensity functions depend on the position of the emitter. We describe the properties of the maximum likelihood and Bayesian estimators. We show that under regularity conditions these estimators are consistent, asymptotically normal and asymptotically efficient. Then we propose some simple consistent estimators and this estimators are further used to construct asymptotically efficient One-step MLE-process.