Poisson source localization on the plane: the smooth case

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.

Poisson Source Localization on the Plane. Change-Point Case

“…As mentioned in the Introduction there are several other statements related to the problem of Poisson source localization depending on the regularity of the signals. The cases of smooth signals and cusp-type signals are considered in the works [3] and [8] respectively. In particular, in [3] the estimation of the parameter ϑ 0 by k ≥ 3 sensors was made in two steps.…”

“…The cases of smooth signals and cusp-type signals are considered in the works [3] and [8] respectively. In particular, in [3] the estimation of the parameter ϑ 0 by k ≥ 3 sensors was made in two steps. First we estimate the moments of the arrival times of the signals, say, τ 1,n , .…”

“…We describe these properties of estimators in the problem of Poisson source localization in the forthcoming work [3].…”

Poisson Source Localization on the Plane. Cusp Case

“…The intensity function λ j,n (ϑ 0 , t) of the Poisson process received by the j-th detector taken in this work and in [1], [4] is of the form λ j,n (ϑ 0 , t) = nλ j (t − τ j ) 1I {t≥τ j } + nλ 0 , 0 ≤ t ≤ T.…”

“…To obtain small errors and have possibility to calculate it we have to consider one or another type of asymptotics. That is why we introduce the large parameter n in the intensity function (1) and study the errors of estimation in the asymptotics n → ∞. This means that the signal and noise are sufficiently large and the estimators θ = θn take values not too far from the true value:…”