Poisson source localization on the plane: the smooth case

• Published in: Metrika Vol. 83; no. 4; pp. 411 - 435
• Main Authors: Chernoyarov, O. V., Kutoyants, Yu. A.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-05-2020; Springer Nature B.V
• Subjects: Asymptotic properties; Economic Theory/Quantitative Economics/Mathematical Methods; Localization; Mathematics and Statistics; Maximum likelihood estimators; Minimax technique; Monte Carlo simulation; Probability Theory and Stochastic Processes; Statistics; Maximum likelihood estimator; Inhomogeneous Poisson process; Source localization; Bayes estimators; Sensors; GPS-localization; One-step MLE
• ISSN: 0026-1335; 1435-926X
• DOI: 10.1007/s00184-019-00738-1

We consider the problem of localization of a Poisson source using observations of inhomogeneous Poisson processes. We assume that
k
detectors are distributed on the plane and each detector generates observations of the Poisson processes, whose intensity functions depend on the position of the source. We study asymptotic properties of the maximum likelihood and Bayesian estimators of the source position on the plane assuming that the amplitude of the intensity functions are large. We show that under regularity conditions these estimators are consistent, asymptotically normal and asymptotically efficient in the minimax mean-square sense. Then we propose some simple consistent estimators and these estimators are further used to construct asymptotically efficient One-step MLE-process.